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.