Testing the Combination of Inotuzumab Ozogamicin and Lower Dose Chemotherapy Compared to Usual Chemotherapy for Adults With B-Cell Acute Lymphoblastic Leukemia or B-Cell Lymphoblastic Lymphoma

Inclusion Criteria:

* PRE-REGISTRATION ELIGIBILITY CRITERIA (STEP 0)
* Research bone marrow or peripheral blood submission

\* This bone marrow or peripheral blood submission is mandatory prior to registration/randomization as baseline for real-time MRD analysis. 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 obtained as soon after pre-registration as possible
* REGISTRATION INCLUSION CRITERIA (STEP 1)
* Diagnosis of B-cell acute lymphoblastic leukemia (ALL) or lymphoblastic lymphoma (LBL) per World Health Organization (WHO) 2016 criteria. Patients must have \>= 5% blasts in the bone marrow or blood. Patients with lymphoblastic lymphoma (LBL) without measurable marrow involvement (\>= 5% blasts) are not eligible

\* T-cell ALL/LBL, Philadelphia-chromosome positive B-cell (as determined by fluorescence in situ hybridization \[FISH\], cytogenetics, or reverse transcriptase polymerase chain reaction \[RT-PCR\]), and Burkitt's like leukemia/lymphoma (mature B-ALL) are not eligible
* Must be CD22 positive by local assessment (\>= 20% by immunohistochemistry or flow cytometry). Patients are eligible regardless of CD20 status but CD20 expression should be assessed at diagnosis by flow cytometry or immunohistochemistry
* Patients must have \>= 5% blasts in the bone marrow or blood. Patients with lymphoblastic lymphoma (LBL) without marrow involvement (\>= 5% blasts) are not eligible
* No prior chemotherapy for ALL except for hydroxyurea (no limit), steroids limited to 7 days, ATRA (no limit), vincristine (single dose), and/or intra-thecal chemotherapy. Leukapheresis is permitted. Palliative radiation to doses 24 Gy or less is permitted. Patients being treated with chronic steroids for other reasons (autoimmune disorder, etc.) are eligible
* Age \>= 50 years
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2. ECOG 3 permitted if related to disease
* Creatinine =\< 2.0 g/dL
* Total bilirubin =\< 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
* AST / ALT =\< 2.5 x upper limit of normal (ULN)
* Cardiac ejection fraction (as measured by multigated acquisition scan \[MUGA\] or echocardiogram) \> 40%
* No clinically relevant liver disease (such as cirrhosis, active hepatitis, or alcohol use disorder), which in the opinion of the treating physician would make this protocol unreasonably hazardous

* Patients with known hepatitis B virus (HBV) infection are eligible if they are on effective HBV suppressive therapy with undetectable HBV viral load and there is no clinically relevant liver disease present (related or unrelated to HBV-related liver damage)
* Patients with known history of hepatitis C virus (HCV) infection are eligible if they have cleared the infection spontaneously or via eradication therapy (HCV viral load undetectable) and there is no clinically relevant liver disease present (related or unrelated to HCV-related liver damage)
* Women and men of reproductive potential should agree to use an appropriate method of birth control throughout their participation in this study due to the teratogenic potential of the therapy utilized in this trial. Include as applicable: Appropriate methods of birth control include abstinence, oral contraceptives, implantable hormonal contraceptives, or double barrier method (diaphragm plus condom)

Exclusion Criteria:

* Physicians should consider whether any of the following may render the patient inappropriate for this protocol:

* Medical condition such as uncontrolled diabetes mellitus, uncontrolled cardiac disease, and uncontrolled pulmonary disease.
* Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial.
* Patients with a "currently active" second malignancy other than non-melanoma skin cancers, early stage prostate cancer, cervical carcinoma in situ, or other cancer for which standard of care would be observation (not requiring treatment). Patients are not considered to have a "currently active" malignancy if they have completed therapy and are free of disease for \>= 1 year, or if the cancer has been surgically resected and considered cured. Patients with a history of multiple myeloma with absence of serum paraprotein for \>= 1 year are not considered to have a "currently active" malignancy.
* REGISTRATION EXCLUSION CRITERIA (STEP 1)
* Patients with symptomatic central nervous system (CNS) disease are not eligible. CNS assessment is not required for eligibility determination if asymptomatic