Inclusion Criteria:

          -  Pre-registration Eligibility Criteria (Step 0)

          -  Submission of bone marrow aspirate and peripheral blood for MRD analysis is mandatory
             prior to registration; the bone marrow sample should be from the first aspiration
             (i.e. first pull). Aspirate needle should be redirected if needed to get first pull
             bone marrow aspirate. It should be initiated as soon as possible after
             pre-registration. The specimens should be sent to the HEME Biobank.

               -  Lumbar Puncture (Spinal Tap) and Intrathecal Methotrexate:

                    -  Patients may receive the day 1 of course IA dose of intrathecal (IT)
                       methotrexate during the prior-to-registration lumbar puncture (or the venous
                       line placement) to avoid a second lumbar puncture. If the dose is
                       administered prior to registration, then systemic chemotherapy must begin
                       within 7 days of this IT chemotherapy.

          -  Registration Eligibility Criteria (Step 1)

          -  Morphologic diagnosis of precursor B-cell acute lymphoblastic leukemia (ALL) based on
             World Health Organization (WHO) criteria. Patients with Burkitt lymphoma/leukemia are
             not eligible.

          -  CD22-positive disease defined as CD22 expression by >= 20% of lymphoblasts by local
             hematopathology evaluation.

          -  Philadelphia chromosome/BCR-ABL1-negative ALL by cytogenetics, fluorescence in situ
             hybridization (FISH), and/or polymerase chain reaction (PCR). If any test is positive
             for Philadelphia chromosome/BCR-ABL1, then the patient is ineligible.

          -  No active central nervous system (CNS) leukemia (i.e. only CNS-1 disease allowed).
             Active CNS leukemia is defined as morphologic evidence of lymphoblasts in the
             cerebrospinal fluid (CSF), use of CNS-directed local treatment for active disease
             within 28 days prior to registration, symptomatic CNS leukemia (i.e. cranial nerve
             palsies or other significant neurological dysfunction) within the 28 days prior to
             registration, and/or known asymptomatic parenchymal CNS mass lesions; see below for
             additional guidance. Prophylactic intrathecal medication alone is not an exclusion.

               -  Categories of CNS Involvement for CNS Evaluation Prior to Registration:

                    -  CNS 1: CSF has < 5 WBC/uL with cytospin negative for blasts; or >= 10 red
                       blood cell (RBC)/uL with cytospin negative for blasts.

                    -  CNS 2: CSF has < 5 WBC/uL with cytospin positive for blasts; or >= 10 RBC/uL
                       with cytospin positive for blasts; or >= 10 RBC/uL, WBC/uL >= 5 but less
                       than Steinherz/Bleyer algorithm with cytospin positive for blasts (see
                       below).

                    -  CNS 3: CSF has >= 5 WBC/uL with cytospin positive for blasts; or >= 10
                       RBC/uL, >= 5 WBC/uL and positive by Steinherz/Bleyer algorithm (see below);
                       or clinical signs of CNS leukemia (such as facial nerve palsy, brain/eye
                       involvement or hypothalamic syndrome). Steinherz/Bleyer Method of Evaluating
                       Initial Traumatic Lumbar Punctures:

                         -  If the patient has leukemia cells in the peripheral blood and the
                            lumbar puncture is traumatic and contains >= 5 WBC/uL with blasts, the
                            following algorithm should be used to define CNS disease: CSF WBC/CSF
                            RBC > 2 x (Blood WBC/Blood RBC count)

          -  Patients with known or suspected testicular involvement by leukemia are allowed
             provided that the patient receives concomitant scrotal/testicular radiotherapy.

               -  Unilateral or bilateral testicular enlargement should be assessed by ultrasound
                  or other imaging technique. Biopsy is recommended if clinical findings are
                  equivocal or suggestive of hydrocele or a non-leukemic mass, but further
                  assessments are per treating physician discretion.

          -  Not pregnant and not nursing.

               -  This study involves agents that have known genotoxic, mutagenic, and teratogenic
                  effects. Therefore, for women of childbearing potential only, a negative
                  pregnancy test done =< 7 days prior to registration is required.

          -  Eastern Cooperative Oncology Group (ECOG) performance status: 0-2

          -  No unstable cardiac disease such as myocardial infarction, angina pectoris,
             uncontrolled heart failure, or uncontrolled cardiac arrhythmia within 6 months of
             registration.

          -  No impaired cardiac function, defined as left ventricular ejection fraction (LVEF) <
             45% or New York Heart Association (NYHA) stage III or IV congestive heart failure
             (CHF).

          -  Patients with known human immunodeficiency virus (HIV) infection are eligible if they
             have been on effective antiretroviral therapy with an undetectable viral load tested
             within 6 months of registration.

          -  Patients with hepatitis B virus (HBV) are eligible only if they meet all the
             following:

               -  On HBV-suppressive therapy.

               -  No evidence of active virus.

               -  No evidence of HBV-related liver damage.

          -  Patients with hepatitis C virus (HCV) are eligible only if they meet all the
             following:

               -  Successfully completed complete-eradication therapy with undetectable viral load.

               -  No evidence of HCV-related liver damage.

          -  No history of clinically relevant neurologic disorder such as epilepsy, seizure,
             aphasia, stroke, severe brain injury, structural brain abnormality, benign brain
             tumor, dementia, Parkinson's disease, movement disorder, cerebellar disease, or other
             significant CNS abnormalities.

          -  No prior additional malignancy (i.e. in addition to ALL) except adequately treated
             basal- or squamous-cell skin cancer, in situ cervical cancer, stage I or II cancer
             from which the patient is currently in complete remission, or any other cancer from
             which the patient has been disease-free for >= 2 years.

          -  No history of clinically significant ventricular arrhythmia, unexplained non-vasovagal
             syncope, or chronic bradycardic states such as sinoatrial block or higher degree of
             atrioventricular block unless a permanent pacemaker has been implanted.

          -  No history of chronic liver disease, including cirrhosis.

          -  No history of sinusoidal occlusion syndrome/veno-occlusive disease of the liver.

          -  No uncontrolled infection or recent history (within 4 months prior to registration) of
             deep tissue infections such as fasciitis or osteomyelitis.

          -  Total bilirubin, serum =< 1.5 x upper limit of normal (ULN)*

               -  Except in the event of: 1) Gilbert disease, in which case total bilirubin must be
                  =< 2 x ULN, or 2) elevated bilirubin believed by investigator to be due to
                  leukemic infiltration, in which case total bilirubin must be =< 2 x ULN.

          -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 2.5 x ULN

          -  Creatinine, serum =< 1.5 ULN OR creatinine clearance >= 40 mL/min

          -  QT interval by Fridericia's correction formula (QTcF) =< 470 msec

          -  Cohort 1 Patients Only

          -  Age >= 60 years.

          -  No prior treatment for ALL except a single dose of intrathecal chemotherapy,
             corticosteroids, hydroxyurea, and/or leukapheresis to reduce peripheral blast count
             and prevent ALL complications. Allowed therapy may be administered for no more than 14
             days and must be completed >= 24 hours prior to the initiation of protocol therapy.

          -  No plan for allogeneic or autologous hematopoietic cell transplantation (HCT).

          -  Cohort 2 Patients Only:

          -  Age >= 18 years.

          -  Relapsed or refractory disease in salvage 1 or 2.

          -  No isolated extramedullary relapse.

          -  Prior allogeneic HCT permitted.

          -  Patients with prior allogeneic HCT must have completed transplantation >= 4 months
             prior to registration.

          -  Patients with prior allogeneic HCT must have no evidence of graft-versus-host disease
             and must have completed immunosuppressive therapy >= 30 days prior to registration.

          -  Prior treatment with inotuzumab ozogamicin, blinatumomab, other CD22-directed therapy,
             or other CD19-directed therapy is not allowed.

          -  Prior treatment with rituximab must be completed >= 7 days prior to registration.

          -  Prior treatment with other monoclonal antibodies must be completed >= 6 weeks prior to
             registration.

          -  Prior treatment for ALL must be completed >= 14 days prior to registration with the
             following exceptions: intrathecal chemotherapy, hydroxyurea, corticosteroids,
             6-mercaptopurine, methotrexate, vincristine, and/or leukapheresis to reduce
             circulating absolute lymphoblast count to =< 10,000/uL or prevent complications
             related to ALL are allowed but must be completed >= 24 hours prior to the initiation
             of protocol therapy.

          -  Patients should have resolution of any acute non-hematologic toxicities of prior
             therapy to National Cancer Institute (NCI) Common Terminology Criteria for Adverse
             Events (CTCAE) version (v)5.0 grade =< 1.

          -  Peripheral blood absolute lymphoblast count =< 10,000/uL (treatment allowed as above
             to reduce blast count to =< 10,000/uL)