Inotuzumab Ozogamicin in Treating Younger Patients With B-Lymphoblastic Lymphoma or Relapsed or Refractory CD22 Positive B Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must be \>= 1 year and \< 22 years of age at the time of enrollment * Patients must have B-ALL, or previously diagnosed B lymphoblastic lymphoma (B-LL), with \>= 5% (M2 or M3) bone marrow blasts with or without extramedullary disease * NOTE: Relapsed patients previously diagnosed with B-lymphoblastic lymphoma (B-LL) are eligible if they have an M2 or M3 marrow at the time of enrollment on this study * Patients with ALL or B-LL who have M2 morphology must have local confirmatory testing showing \>= 5% blasts by flow cytometry, fluorescence in situ hybridization (FISH) testing or other molecular method * Leukemic blasts must demonstrate surface expression of CD22 at the time of relapse by local/institutional flow cytometry of a bone marrow aspirate sample; (assessment of CD22 using a bright fluorophore such as phycoerythrin \[PE\] is strongly recommended) * In the case of an inadequate aspirate sample (dry tap) or if bone marrow aspirate is unable to be performed due to patient clinical status, flow cytometry of peripheral blood specimen may be substituted if the patient has at least 1,000/uL circulating blasts; alternatively, CD22 expression may be documented by immunohistochemistry of a bone marrow biopsy specimen * Patients with one of the following: * Second or greater relapse; * Primary refractory disease with at least 2 prior induction attempts; * First relapse refractory to at least one prior re-induction attempt * Any relapse after HSCT (Cohort 1 ONLY) Patients with Down syndrome are eligible ONLY for Cohort 1 with: * Any of above disease status, OR * First relapse with no prior re-induction attempt NOTE: Patients with Down syndrome or prior HSCT are NOT eligible for Cohort 2 combination therapy * Patients with Philadelphia chromosome (Ph)+ ALL must have had two prior therapy attempts including two different tyrosine kinase inhibitors (TKIs) * Patients must have fully recovered from the acute toxic effects of all prior anti-cancer therapy, defined as resolution of all such toxicities to =\< grade 2 or lower per the inclusion/exclusion criteria prior to entering this study. Apply to Cohort 2: * Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive. For agents not listed, the duration of this interval must be discussed with the study chair and the study-assigned Research Coordinator prior to enrollment. * A waiting period prior to enrollment is not required for patients receiving standard cytotoxic maintenance chemotherapy (i.e., corticosteroid, vincristine, 6MP, and/or methotrexate). * A waiting period is not required for patients receiving a single dose of intrathecal methotrexate, hydrocortisone, and/or cytarabine within 7 days prior to enrollment * \>= 14 days must have elapsed after the completion of other cytotoxic therapy, with the exception of hydroxyurea, for patients not receiving standard maintenance therapy. For patients who previously received calaspargase pegol, \>= 21 days must have elapsed after the last dose. Additionally, patients must have fully recovered from all acute toxic effects of prior therapy. * Note: Cytoreduction with hydroxyurea must be discontinued \>= 24 hours prior to the start of protocol therapy. * Anti-cancer agents not known to be myelosuppressive (e.g., not associated with reduced platelet or absolute neutrophil count \[ANC\] counts): \>= 7 days after the last dose of agent. For agents not listed, the duration of this interval must be discussed with the study chair and the study-assigned research coordinator prior to enrollment. * Anti-cancer agents that are antibodies: \>= 21 days must have elapsed from infusion of last dose of antibody, and toxicity related to prior antibody therapy must be recovered to grade =\< 1. There is an exception for blinatumomab infusions, for which patients must have been off for at least 3 days and all drug related toxicity must have resolved to grade 2 or lower as outlined in the inclusion/exclusion criteria. * Corticosteroids: If used to modify immune adverse events related to prior therapy, \>= 14 days must have elapsed since last dose of corticosteroid. A waiting period prior to enrollment is not required for patients receiving corticosteroid for leukemia therapy/cytoreduction. * Radiotherapy: \>= 2 weeks must have elapsed since local palliative radiation therapy (XRT) (small port); \>= 3 months must have elapsed if prior cranial or craniospinal XRT was received, if \>= 50% of the pelvis was irradiated, or if total body irradiation (TBI) was received; \>= 6 weeks must have elapsed if other substantial bone marrow irradiation was given. * Stem cell transplant or rescue without TBI: For Cohort 1, at least 90 days must have elapsed since stem cell transplant and at least 30 days from donor lymphocyte infusion. Patient must have had no more than one previous HSCT and currently have no evidence of active graft vs. host disease (GVHD). For Cohort 2, no prior HSCT is allowed. * Chimeric antigen receptor (CAR) T cell therapy: At least 30 days must have elapsed from the last CAR-T cell infusion * Patients must have a performance status corresponding to Eastern Cooperative Oncology Group (ECOG) scores of 0, 1, or 2; use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score * Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 70 mL/min/1.73 m\^2 or * A serum creatinine based on age/sex as follows: * 1 to \< 2 years: maximum serum creatinine 0.6 mg/dL (both male and female) * 2 to \< 6 years: maximum serum creatinine 0.8 mg/dL (both male and female) * 6 to \< 10 years: maximum serum creatinine 1 mg/dL (both male and female) * 10 to \< 13 years: maximum serum creatinine 1.2 mg/dL (both male and female) * 13 to \< 16 years: maximum serum creatinine 1.5 mg/dL (male), 1.4 mg/dL (female) * \>= 16 years: maximum serum creatinine 1.7 mg/dL (male), 1.4 mg/dL (female) * Direct bilirubin =\< 1.5 x upper limit of normal (ULN) for age, and * Serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 5 x ULN for age; for the purpose of this study, the ULN for ALT will be 45 U/L Exclusion Criteria: * Patients with any prior history of SOS irrespective of severity * Patients with isolated central nervous system (CNS), testicular, or any other extramedullary site of relapse * Patients who have been previously treated with inotuzumab ozogamicin * Patients who have previously received HSCT (Cohort 2 only) * Patients with Down syndrome (Cohort 2 only) * History of allergic reaction attributed to compounds of similar or biologic composition to inotuzumab ozogamicin or other agents in the study * Note: Patients with history of allergy to pegaspargase/calaspargase pegol are eligible for enrollment on Cohort 2 if Erwinia formulation of asparaginase can be obtained * Patients with active optic nerve and/or retinal involvement are not eligible; patients who are presenting with visual disturbances should have an ophthalmologic exam and, if indicated, a magnetic resonance imaging (MRI) to assess optic nerve or retinal involvement * Patients who are currently receiving another investigational drug * Patients who are currently receiving or plan to receive other anti-cancer agents (except hydroxyurea, which may be continued until 24 hours prior to start of protocol therapy, and intrathecal chemotherapy) * Anti-GVHD or agents to prevent organ rejection post-transplant; patients who are receiving cyclosporine, tacrolimus, or other agents to prevent either graft-versus-host disease post bone marrow transplant or organ rejection post-transplant are not eligible for this trial; at least 3 half-lives must have elapsed after the last dose of GVHD or anti-rejection medications * Patients who are currently receiving or plan to receive corticosteroids except as described below * Systemic corticosteroids may be administered for cytoreduction up to 24 hours prior to the start of protocol therapy, (Cohort 1 only) for all patients, corticosteroids may be administered as a premedication for inotuzumab ozogamicin and as treatment for allergic reactions or for physiologic replacement/stress dosing of hydrocortisone for documented adrenal insufficiency; corticosteroids are not allowed for other indications * Patients with known human immunodeficiency virus (HIV), hepatitis B or C infections; testing to prove negative status is not required for enrollment unless it is deemed necessary for usual medical care of the patient * Patients who have an active uncontrolled infection defined as: * Positive bacterial blood culture within 48 hours of study enrollment; * Fever above 38.2 degree Celsius (C) within 48 hours of study enrollment with clinical signs of infection; fever that is determined to be due to tumor burden is allowed if patients have documented negative blood cultures for at least 48 hours prior to enrollment and no concurrent signs or symptoms of active infection or hemodynamic instability * A positive fungal culture within 30 days of study enrollment or active therapy for presumed invasive fungal infection * Patients may be receiving IV or oral antibiotics to complete a course of therapy for a prior documented infection as long as cultures have been negative for at least 48 hours and signs or symptoms of active infection have resolved; for patients with clostridium (C.) difficile diarrhea, at least 72 hours of antibacterial therapy must have elapsed and stools must have normalized to baseline * Active viral or protozoal infection requiring IV treatment * Patients known to have one of the following concomitant genetic syndromes: Bloom syndrome, ataxia-telangiectasia, Fanconi anemia, Kostmann syndrome, Schwachman (Schwachman-Diamond-Blackfan) syndrome or any other known bone marrow failure syndrome * There have been no human studies of inotuzumab ozogamicin in pregnant women and no reports of exposure in utero; based on nonclinical safety studies, inotuzumab ozogamicin has the potential to impair human male and female fertility and to adversely affect human embryo fetal development; women of childbearing potential should be advised to avoid becoming pregnant while receiving inotuzumab ozogamicin; there is no information regarding the presence of inotuzumab ozogamicin in human milk, the effects on the breast-fed infant, or the effects on milk production; because of the potential for adverse reactions in breast-fed infants, women should not breast-feed during treatment with inotuzumab ozogamicin and for at least 2 months after the final dose * Female patients of childbearing potential are not eligible unless a negative pregnancy test result has been obtained within 7 days prior to enrollment * Female patients who are sexually active and of reproductive potential are not eligible unless they agree to use an effective contraceptive method for the duration of their study participation and for 8 months after the last dose of inotuzumab ozogamicin * Men with female partners of childbearing potential should use effective contraception during treatment with inotuzumab ozogamicin and for at least 5 months after the last dose of inotuzumab ozogamicin * Lactating females are not eligible unless they agree not to breastfeed their infants