S1312, Inotuzumab Ozogamicin and Combination Chemotherapy in Treating Patients With Relapsed or Refractory Acute Leukemia

Inclusion Criteria: * Patients must have a diagnosis of relapsed or refractory CD22-positive acute leukemia including B-ALL, mixed phenotype leukemia (biphenotypic), or Burkitt's leukemia based on World Health Organization (WHO) classification; patients with bilineal leukemia are excluded * Patients must have evidence of acute leukemia in their peripheral blood or bone marrow; patients must have \>= 5% blasts in the peripheral blood or bone marrow within 14 days prior to registration; at least \>= 20% of those blasts must be CD22-positive (surface) based on local immunophenotyping and histopathology * Patients must be refractory or have relapsed following prior induction therapy; a standard induction regimen is defined as any program of treatment that includes vincristine and prednisone or dexamethasone, cytarabine/anthracycline, or high dose cytarabine * For sites with the B1931022 pharmaceutical trial open, precursor B-cell ALL patients from that site may be eligible for S1312 providing they meet the following criteria: * Patient is in second salvage or more; OR * Patient was treated on the standard of care arm of B1931022 and failed therapy * Patients may have received prior allogeneic transplant or autologous transplant; however, patients with prior allogeneic bone marrow transplant will be eligible only if both of the following conditions are met: * The transplant must have been performed \>= 90 days prior to registration * The patient must not have \>= grade 2 acute graft versus host disease (GvHD) or either moderate or severe limited chronic GvHD within 14 days prior to registration * Patients known to have Philadelphia chromosome positive (Ph+) ALL must have either failed treatment or been intolerant to treatment with at least two second or third generation tyrosine kinase inhibitors * Patients must not have received prior treatment with inotuzumab ozogamicin; previous treatment with other anti-CD22 antibodies must have been completed at least 90 days prior to registration * Patients must have Zubrod performance status 0-2 * Patients must not have received any chemotherapy, investigational agents, or undergone major surgery within 14 days prior to registration with the following exceptions: * Monoclonal antibodies must not have been received for 1 week prior to registration * Chimeric antigen receptor (CAR) T-cells must not have been received for 28 days prior to registration. * Steroids, hydroxyurea, vincristine, 6-mercaptopurine, methotrexate, thioguanine and intrathecal chemotherapy are permitted within any time frame prior to registration. FDA-approved TKIs may also be administered until 1 day prior to start of study therapy (C1, D1). * All drug-related toxicities must have resolved to =\< grade 2 * Patients must not have a systemic bacterial, fungal, or viral infection that is not controlled (defined as exhibiting ongoing signs/symptoms related to the infection and without improvement despite appropriate antibiotics or other treatment) * Patients must not have any other serious concurrent disease or have a history of serious organ dysfunction or disease involving the heart, kidney, liver, or other organ system that would put the patient at undue risk of undergoing therapy * Patients must not have active central nervous system (CNS) involvement (by clinical evaluation); patients with previous documented history of CNS involvement of acute leukemia, or with clinical signs or symptoms consistent with CNS involvement of acute leukemia, must have a lumbar puncture which is negative for CNS involvement of acute leukemia; the lumbar puncture must be completed within 14 days prior to registration; patients with no previous history of documented CNS involvement and with no clinical signs or symptoms consistent with CNS involvement are not required to have completed a lumbar puncture before registration; note that treatment with intrathecal therapy is recommended during protocol treatment but CNS analysis during treatment is not required * Patients must have a peripheral blast count \< 25,000/uL within 2 days prior to registration; (treatment with hydroxyurea and steroids is permitted to bring the countdown) * Patients must have serum creatinine =\< 2 x institutional upper limits of normal (IULN) within 7 days prior to registration * Patients must have bilirubin =\< 2 x IULN within 7 days prior to registration (unless the bilirubin is primarily unconjugated) * Patients must have \< grade 2 neuropathy (sensory/motor) within 7 days prior to registration * Patients must have serum glutamic oxaloacetic transaminase (SGOT) and serum glutamate pyruvate transaminase (SGPT) =\< 2.5 x IULN within 7 days prior to registration * Patients with a history of a serious allergic or anaphylactic reaction to humanized monoclonal antibodies are not eligible * Patients must not have a history of chronic or active hepatitis B or C infection; patients must have negative hepatitis B and C serologies performed within 28 days prior to registration * Patients must not have evidence or history of veno-occlusive disease or sinusoidal obstruction syndrome * Patients must not have a cardiac ejection fraction \< 45% or the presence of New York Heart Association stage III or IV heart failure within 14 days prior to registration; either echocardiogram (ECHO) or multiple-gated acquisition scan (MUGA) may be used to determine ejection fraction * Patients must not have a myocardial infarction within 6 months prior to registration * Patients must not have a history of clinically significant arrhythmia, prolonged corrected QT (QTc) interval, or unexplained syncope not thought to be vasovagal in nature within 6 months prior to registration * Patients must not have a screening corrected QT using Fridericia's formula (QTcF) interval \> 500 milliseconds (by Fridericia calculation) based on the average of triplicate electrocardiogram (EKG) performed within 7 days prior to registration; note that triplicate EKG is required at other timepoints * Patients must not have a history of chronic liver disease (or cirrhosis) * Patients who are known to be human immunodeficiency virus (HIV)+ are eligible providing they meet all of the following additional criteria within 28 days prior to registration: * CD4+ cells \>= 350/mm\^3 (nadir) * Viral load of \< 50 copies HIV messenger ribonucleic acid (mRNA)/mm\^3 if on combination antiretroviral therapy (cART) or \< 25,000 copies HIV mRNA if not on cART * No zidovudine or stavudine as part of cART Patients who are HIV+ and do not meet all of these criteria are not eligible for this study * Patients with evidence of extramedullary disease at diagnosis will have computed tomography (CT) scan of the chest, abdomen and pelvis to obtain baseline values within 28 days prior to registration * Patients must have complete history and physical examination within 28 days prior to registration * Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures * Prior malignancy other than acute leukemia is allowed, provided it is in remission and there is no plan to treat the malignancy at the time of registration * Pretreatment cytogenetics must be performed on all patients; collection of pretreatment specimens must be completed within 14 days prior to registration to S1312; specimens must be submitted to the site's preferred Clinical Laboratory Improvement Amendments (CLIA)-approved cytogenetics laboratory; reports of the results must be submitted as described; note that cytogenetics are required at other time points * Patients or their legally authorized representative must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * As a part of the OPEN registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system * Patients planning to enroll in this study must first have a slot reserved in advance of the registration; all site staff will use OPEN to create a slot reservation