This phase I trial studies the best dose of inotuzumab ozogamicin in combination with chemotherapy in treating patients with B-cell acute lymphoblastic leukemia that has come back (recurrent) or that does not respond to treatment (refractory). Inotuzumab ozogamicin is a monoclonal antibody, called inotuzumab, linked to a toxic agent called ozogamicin. Inotuzumab attaches to CD22 positive cancer cells in a targeted way and delivers ozogamicin to kill them. Drugs used in chemotherapy, such as etoposide, prednisone, vincristine, cyclophosphamide, and doxorubicin, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving inotuzumab ozogamicin in combination with chemotherapy may kill more cancer cells than with chemotherapy alone in treating patients with recurrent or refractory B-cell acute lymphoblastic leukemia.