Inclusion Criteria:

          -  Patients must have B-ALL, or previously diagnosed B lymphoblastic lymphoma (B-LL),
             with >= 5% (M2 or M3) bone marrow blasts with or without extramedullary disease

               -  NOTE: Relapsed patients previously diagnosed with B-lymphoblastic lymphoma (B-LL)
                  are eligible if they have an M2 or M3 marrow at the time of enrollment on this
                  study

          -  Patients with ALL or B-LL who have M2 morphology must have local confirmatory testing
             showing >= 5% blasts by flow cytometry, fluorescence in situ hybridization (FISH)
             testing or other molecular method

          -  Leukemic blasts must demonstrate surface expression of CD22 at the time of relapse by
             local/institutional flow cytometry of a bone marrow aspirate sample; (assessment of
             CD22 using a bright fluorophore such as phycoerythrin [PE] is strongly recommended)

               -  In the case of an inadequate aspirate sample (dry tap) or if bone marrow aspirate
                  is unable to be performed due to patient clinical status, flow cytometry of
                  peripheral blood specimen may be substituted if the patient has at least 1,000/uL
                  circulating blasts; alternatively, CD22 expression may be documented by
                  immunohistochemistry of a bone marrow biopsy specimen

          -  Patients with one of the following:

               -  Second or greater relapse;

               -  Primary refractory disease with at least 2 prior induction attempts;

               -  First relapse refractory to at least one prior re-induction attempt

               -  Any relapse after HSCT (Cohort 1 ONLY)

        Patients with Down syndrome are eligible ONLY for Cohort 1 with:

          -  Any of above disease status, OR

          -  First relapse with no prior re-induction attempt NOTE: Patients with Down syndrome or
             prior HSCT are NOT eligible for Cohort 2 combination therapy

               -  Patients with Philadelphia chromosome (Ph)+ ALL must have had two prior therapy
                  attempts including two different tyrosine kinase inhibitors (TKIs)

               -  Patients must have fully recovered from the acute non-hematologic toxic effects
                  of all prior anti-cancer therapy, defined as resolution of all such toxicities to
                  =< grade 2 or lower per the inclusion/exclusion criteria prior to entering this
                  study

          -  Myelosuppressive chemotherapy:

               -  No waiting period will be required for patients receiving standard
                  "maintenance-like" chemotherapy including oral mercaptopurine, weekly low-dose
                  oral methotrexate, and intermittent vincristine/steroid pulses; otherwise, at
                  least 14 days must have elapsed since the completion of cytotoxic therapy, with
                  the exceptions of hydroxyurea or corticosteroids used for cytoreduction

               -  Intrathecal cytotoxic therapy: No waiting period is required for patients having
                  received intrathecal cytarabine, methotrexate, and/or hydrocortisone; intrathecal
                  chemotherapy given at the time of diagnostic lumbar puncture (LP) to evaluate for
                  relapse prior to study enrollment is allowed

          -  At least 7 days must have elapsed since the completion of therapy with a growth
             factor; at least 14 days must have elapsed after receiving pegfilgrastim

          -  At least 7 days must have elapsed since completion of therapy with a biologic agent
             (including tyrosine kinase inhibitors); for agents that have known adverse events
             occurring beyond 7 days after administration, this period prior to enrollment must be
             extended beyond the time during which adverse events are known to occur

          -  At least 3 half-lives must have elapsed since prior therapy that included a monoclonal
             antibody with the exception of blinatumomab; patients must have been off blinatumomab
             infusion for at least 3 days and all drug related toxicity must have resolved to grade
             2 or lower as outlined in the inclusion/exclusion criteria

          -  >= 2 weeks must have elapsed since local palliative radiation therapy (XRT) (small
             port); >= 3 months must have elapsed if prior cranial or craniospinal XRT was
             received, if >= 50% of the pelvis was irradiated, or if total-body irradiation (TBI)
             was received; >= 6 weeks must have elapsed if other substantial bone marrow
             irradiation was given

          -  For Cohort 1, at least 90 days must have elapsed since stem cell transplant and at
             least 30 days from donor lymphocyte infusion; patient must have had no more than one
             previous HSCT and currently have no evidence of active graft versus (vs.) host disease
             (GVHD); for Cohort 2, no prior HSCT is allowed

          -  At least 30 days must have elapsed from the last chimeric antigen receptor (CAR)-T
             cell infusion

               -  Patients must have a performance status corresponding to Eastern Cooperative
                  Oncology Group (ECOG) scores of 0, 1, or 2; use Karnofsky for patients > 16 years
                  of age and Lansky for patients =< 16 years of age; patients who are unable to
                  walk because of paralysis, but who are up in a wheelchair, will be considered
                  ambulatory for the purpose of assessing the performance score

               -  Creatinine clearance or radioisotope glomerular filtration rate (GFR) >= 70
                  mL/min/1.73 m^2 or

               -  A serum creatinine based on age/gender as follows:

          -  1 to < 2 years: maximum serum creatinine 0.6 mg/dL (both male and female)

          -  2 to < 6 years: maximum serum creatinine 0.8 mg/dL (both male and female)

          -  6 to < 10 years: maximum serum creatinine 1 mg/dL (both male and female)

          -  10 to < 13 years: maximum serum creatinine 1.2 mg/dL (both male and female)

          -  13 to < 16 years: maximum serum creatinine 1.5 mg/dL (male), 1.4 mg/dL (female)

          -  >= 16 years: maximum serum creatinine 1.7 mg/dL (male), 1.4 mg/dL (female)

               -  Direct bilirubin =< 1.5 x upper limit of normal (ULN) for age, and

               -  Serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase [ALT]) =< 5
                  x ULN for age; for the purpose of this study, the ULN for ALT will be 45 U/L

        Exclusion Criteria:

          -  Patients with any prior history of SOS irrespective of severity

          -  Patients with isolated central nervous system (CNS), testicular, or any other
             extramedullary site of relapse

          -  Patients who have been previously treated with inotuzumab ozogamicin

          -  Patients who have previously received HSCT (Cohort 2 only)

          -  Patients with Down syndrome (Cohort 2 only)

          -  History of allergic reaction attributed to compounds of similar or biologic
             composition to inotuzumab ozogamicin or other agents in the study

               -  Note: Patients with history of allergy to pegaspargase are eligible for
                  enrollment on Cohort 2 (dose levels 1 and -1) if asparaginase Erwinia can be
                  obtained

               -  If Cohort 2 is enrolling at dose level -2, then patients who cannot receive
                  asparaginase due to prior allergy, toxicity, or lack of access may enroll

               -  NOTE: patients on AALL1621 are not eligible to co-enroll on AALL1931

          -  Patients with active optic nerve and/or retinal involvement are not eligible; patients
             who are presenting with visual disturbances should have an ophthalmologic exam and, if
             indicated, a magnetic resonance imaging (MRI) to assess optic nerve or retinal
             involvement

          -  Patients who are currently receiving another investigational drug

          -  Patients who are currently receiving or plan to receive other anti-cancer agents
             (except hydroxyurea, which may be continued until 24 hours prior to start of protocol
             therapy and intrathecal chemotherapy)

          -  Anti-GVHD or agents to prevent organ rejection post-transplant; patients who are
             receiving cyclosporine, tacrolimus, or other agents to prevent either
             graft-versus-host disease post bone marrow transplant or organ rejection
             post-transplant are not eligible for this trial; at least 3 half-lives must have
             elapsed after the last dose of GVHD medications (meds)

          -  Patients who are currently receiving or plan to receive corticosteroids except as
             described below

               -  Systemic corticosteroids may be administered for cytoreduction up to 24 hours
                  prior to the start of protocol therapy, (Cohort 1 only) for all patients,
                  corticosteroids may be administered as a premedication for inotuzumab ozogamicin
                  and as treatment for allergic reactions or for physiologic replacement/stress
                  dosing of hydrocortisone for documented adrenal insufficiency; corticosteroids
                  are not allowed for other indications

          -  Patients with known human immunodeficiency virus (HIV), hepatitis B or C infections;
             testing to prove negative status is not required for enrollment unless it is deemed
             necessary for usual medical care of the patient

          -  Patients who have an active uncontrolled infection defined as:

               -  Positive bacterial blood culture within 48 hours of study enrollment;

               -  Fever above 38.2 degree Celsius (C) within 48 hours of study enrollment with
                  clinical signs of infection; fever that is determined to be due to tumor burden
                  is allowed if patients have documented negative blood cultures for at least 48
                  hours prior to enrollment and no concurrent signs or symptoms of active infection
                  or hemodynamic instability

               -  A positive fungal culture within 30 days of study enrollment or active therapy
                  for presumed invasive fungal infection

               -  Patients may be receiving IV or oral antibiotics to complete a course of therapy
                  for a prior documented infection as long as cultures have been negative for at
                  least 48 hours and signs or symptoms of active infection have resolved; for
                  patients with clostridium (C.) difficile diarrhea, at least 72 hours of
                  antibacterial therapy must have elapsed and stools must have normalized to
                  baseline

               -  Active viral or protozoal infection requiring IV treatment

          -  Patients known to have one of the following concomitant genetic syndromes: Bloom
             syndrome, ataxia-telangiectasia, Fanconi anemia, Kostmann syndrome, Shwachmann
             (Shwachmann-Diamond-Blackfan) syndrome or any other known bone marrow failure syndrome

          -  There have been no human studies of inotuzumab ozogamicin in pregnant women and no
             reports of exposure in utero; based on nonclinical safety studies, inotuzumab
             ozogamicin has the potential to impair human male and female fertility and to
             adversely affect human embryo fetal development; women of childbearing potential
             should be advised to avoid becoming pregnant while receiving inotuzumab ozogamicin;
             there is no information regarding the presence of inotuzumab ozogamicin in human milk,
             the effects on the breast-fed infant, or the effects on milk production; because of
             the potential for adverse reactions in breast-fed infants, women should not
             breast-feed during treatment with inotuzumab ozogamicin and for at least 2 months
             after the final dose

               -  Female patients of childbearing potential are not eligible unless a negative
                  pregnancy test result has been obtained within 7 days of starting protocol
                  therapy

               -  Female patients who are sexually active and of reproductive potential are not
                  eligible unless they agree to use an effective contraceptive method for the
                  duration of their study participation and for 8 months after the last dose of
                  inotuzumab ozogamicin

               -  Men with female partners of childbearing potential should use effective
                  contraception during treatment with inotuzumab ozogamicin and for at least 5
                  months after the last dose of inotuzumab ozogamicin

               -  Lactating females are not eligible unless they agree not to breastfeed their
                  infants