Inotuzumab Ozogamicin and Chemotherapy in Treating Patients With Recurrent or Refractory B-cell Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must have a confirmed diagnosis of CD22-positive, Philadelphia chromosome (Ph)-positive or Ph-negative B-cell acute lymphoblastic leukemia or lymphoblastic lymphoma. CD22 expression will be determined by multiparameter flow cytometry (MFC) or immunohistochemistry. * Relapsed or refractory disease, as defined by any of the following: * Unable to achieve complete response (CR) despite \>= 4 weeks of initial course of systemic therapy. * Recurrence of disease at any point after CR was achieved. * (Note: patients with Ph-positive disease must have received \>= 1 second- or third-generation ABL kinase inhibitor as part of their prior treatment to be eligible.) * Detectable disease, as defined by any of the following: * Presence of \>= 5% abnormal blasts in the bone marrow or peripheral blood by morphology or MFC. * Patients with isolated extramedullary disease will be permitted if there is \>= 1 site of disease that measures \>= 1.5 cm in longest diameter on cross-sectional imaging. * Absolute neutrophil count (ANC) \>= 1,000/uL. * Hemoglobin \>= 8 g/dL. * Platelets \>= 50,000/uL. * Note: Transfusions and growth factor support will be permitted within 3 days of initiation of study treatment to reach these thresholds. As patients with relapsed/refractory ALL frequently have cytopenias due to marrow infiltration by the disease, no hematologic parameters will be required for enrollment if cytopenias can be attributed to disease. * Total serum bilirubin =\< 1.5 x upper limit of normal (ULN); (unless due to Gilbert syndrome or hemolysis; =\< 2 x ULN for hepatic abnormalities considered disease-related). * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN. * Serum creatinine =\< 1.5 x ULN or serum creatinine level associated with a measured or calculated creatinine clearance of \>= 40 mL/min. * Corrected QT (QTc) interval =\< 500 msec; if assessed, left ventricular ejection fraction \>= 40%. * An Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2. * Must agree to the use of effective contraception while on study treatment, unless they are highly unlikely to conceive (defined as \[1\] surgically sterilized, or \[2\] women who are and men whose sexual partner\[s\] is/are postmenopausal \[i.e., a woman who is \> 50 years old or who has not had menses for \>= 1 year\], or \[3\] not heterosexually active for the duration of the study). Exclusion Criteria: * Patients with a circulating blast count of \> 50,000/uL; systemic therapy with either hydroxyurea, vincristine, and/or corticosteroids will be permitted within 3 days of initiation of study treatment to reduce the blast count. * Except for management of circulating blasts noted above, adequate duration from prior therapy must be achieved before initiation of study treatment, as defined below: * No cytotoxic or targeted systemic therapy \< 2 weeks or 5 half-lives (whichever is shorter). * No blinatumomab \< 2 weeks. * No radiation therapy \< 4 weeks. * No monoclonal antibody therapy \< 6 weeks (except for prior InO, as discussed below). * Patients previously treated with InO will be eligible, unless they meet ANY of the following criteria: * \> 6 individual doses (e.g., \> 2 standard cycles) were administered. * Any documented hepatic toxicity observed was grade 3 or higher. * The most recent dose was administered \< 3 months from the initiation of study treatment. * For patients that have received prior allogeneic HCT, they must be \>= 4 months from the date of stem cell infusion, with no prior history of sinusoidal obstruction syndrome/veno-occlusive disease (SOS/VOD), and off all treatment for graft-vs-host disease (GVHD) for \>= 2 weeks. Patients with minimal active symptoms that can be controlled with topical therapies and/or the equivalent of prednisone =\< 10 mg/day will be eligible. * For patients that have received other forms of cellular immunotherapy (e.g., chimeric antigen receptor-modified \[CAR\] T cells), they must be \>= 21 days from cell infusion, and any specific manifestations of cytokine release syndrome or neurologic toxicity attributable to the cellular therapy have completely resolved (i.e., \< grade 1) * Patients with a known history of chronic liver disease, including but not limited to cirrhosis, steatohepatitis, and chronic viral hepatitis. Patients with a history of GVHD of the liver will be permitted, provided they meet all of the other eligibility criteria. * Patients with isolated testicular or central nervous system disease. * Known hypersensitivity or intolerance to any of the agents under investigation. * May not be pregnant or nursing.