Inotuzumab Ozogamicin and Vincristine Sulfate Liposome in Treating Patients With Relapsed or Refractory CD22+ B-cell Acute Lymphoblastic Leukemia

Inclusion Criteria:

* Eastern Cooperative Oncology Group performance status between 0-2
* Serum creatinine =\< 1.5 x upper limit of normal (ULN) or calculated creatinine clearance (Cockcroft and Gault) \> 30 mL/min
* Alanine aminotransferase (ALT) =\< 5 x ULN
* Total bilirubin= \< 1.5 x ULN
* Left ventricular ejection fraction (LVEF) \>= 40% as assessed by echocardiogram (ECHO) or multiple-gated acquisition (MUGA) scan performed within 28 days of enrollment
* Diagnosis of relapsed/refractory CD22+ B-cell ALL with disease in the bone marrow and/or peripheral blood by morphology (\>=5% blasts). CD22-positive B-ALL is defined as expression by at least 20% of malignant lymphoblasts as determined by local flow cytometry and/or immunohistochemistry from a peripheral blood and/or bone marrow sample obtained within 2 weeks of screening
* Relapsed or refractory disease, defined as second or greater bone marrow relapse from CR or overall response or, disease has progressed following two or more anti-leukemia therapies. Specifically:

  * Any bone marrow relapse after allogeneic HSCT: subjects must be at least 1 month from HSCT at the time of screening and off immunosuppressive medication for at least 2 weeks at time of initial treatment (with the exception of low-dose steroids =\< 20 mg prednisone equivalent) and have no active graft versus (vs.) host disease (GVHD);
  * Philadelphia chromosome (Ph) negative B-ALL which has not achieved CR or CRi after at least 2 attempts at remission induction using standard intensive chemotherapy regimen(s);
  * Philadelphia chromosome (Ph) positive B-cell ALL intolerant to or ineligible for BCR-ABL tyrosine kinase inhibitor (TKI) therapy or with disease which has progressed after at least two lines of prior TKI therapy
* Participants of childbearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry and for females, at least 8 months after the final dose of inotuzumab ozogamicin. Males with female partners of childbearing potential must agree to use adequate contraceptive prior to study entry and for at least 5 months after the final dose of inotuzumab ozogamicin. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure

Exclusion Criteria:

* Receipt of chemotherapy, radiotherapy, or investigational drug therapy within 2 weeks prior to treatment on study or those who have not recovered from adverse events due to agents administered \> 2 weeks earlier
* Participants on oral or injectable calcineurin inhibitors (e.g., cyclosporin, tacrolimus) within 4 weeks prior to study enrollment

  * Active central nervous system involvement; patients who have a history of central nervous system (CNS) disease which has been effectively treated (as defined by at least one negative cerebrospinal sample prior to screening) are eligible
* Prior malignancy, unless treated with curative intent and with no evidence of active disease present for \> 5 years before screening, with the following exceptions:

  * Subjects with stage I breast cancer that has been completely and successfully treated, requiring no therapy or only anti-hormonal therapy
  * Subjects with T1N0M0 or T2N0M0 colorectal cancer who have been completely and successfully resected and who are disease-free for \> 2 years prior to screening
  * Subjects with indolent prostate cancer, defined as clinical stage T1 or T2a, Gleason score =\< 6, and prostate-specific antigen (PSA) \< 10 ng/mL, requiring no therapy or only anti-hormonal therapy
  * Subjects with a history of basal cell or squamous cell carcinoma of the skin, or carcinoma in situ of the cervix, fully resected, and with no evidence of active disease
* Uncontrolled intercurrent medical illness including, but not limited to, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that in the opinion of the investigator would limit compliance with study requirements
* Uncontrolled systemic fungal, bacterial, viral, or other infection defined as exhibiting ongoing signs and symptoms due to infection despite appropriate anti-infective therapy at time of screening
* Pregnant or nursing female participants
* Known active hepatitis B, known active hepatitis C, or any human immunodeficiency virus (HIV) infection at the time of screening
* Presence of grade II-IV acute or extensive chronic graft versus host disease (GVHD) at time of screening
* Unwilling or unable to follow protocol requirements
* Any condition which in the investigator's opinion deems the participant an unsuitable candidate to receive study drug including, but not limited to, medical, psychological, familial, social, or geographical considerations