Inotuzumab Ozogamicin in Treating Patients With Relapsed or Refractory CD22 Positive Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients at least 12 years of age * Patients with a diagnosis of CD22-positive acute lymphoblastic leukemia (ALL) based on local immunophenotyping and histopathology who have: * Refractory disease, defined as disease progression or no response while receiving their most recent prior anti-cancer therapy, * Relapsed disease, defined as response to their most recent prior anti-cancer therapy with subsequent relapse * Performance status of 0 to 3 * Serum creatinine =\< 2 x upper limit of normal (ULN) or estimated creatinine clearance \>= 15 mL/min as calculated using the method standard for the institution * Total serum bilirubin =\< 1.5 x ULN unless the patient has documented Gilbert syndrome. If organ function abnormalities are considered due to tumor, total serum bilirubin must be =\< 2 x ULN * Aspartate and alanine aminotransferase (AST or ALT) =\< 2.5 x ULN * No active or co-existing malignancy requiring chemotherapy or radiation within 6 months * Female subjects of childbearing potential should be willing to use effective methods birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. Effective methods of birth control include birth control pills or injections, intrauterine devices (IUDs), or double-barrier methods (for example, a condom in combination with spermicide) * Male subjects should agree to use an effective method of contraception starting with the first dose of study therapy through the duration of treatment Exclusion Criteria: * Pregnant or nursing women * Known to be human immunodeficiency virus (HIV)+ * Philadelphia chromosome (Ph)+ ALL * Active and uncontrolled disease/infection as judged by the treating physician * Unable or unwilling to sign the consent form * Prior allogeneic stem cell transplantation (ASCT) or other anti-CD22 immunotherapy within =\< 4 months before first dose of study treatment * Active central nervous system (CNS) or extramedullary disease unless approved by the principal investigator (PI) * Monoclonal antibodies therapy within 2 weeks before study entry * Radiotherapy and cancer chemotherapy (except for intrathecal chemotherapy, hydroxyurea, and cytarabine. Cytarabine and hydroxyurea are allowed to be used emergently in case of leukocytosis) or any investigational drug within 2 weeks before study entry * Evidence or history of veno-occlusive disease (VOD) or sinusoidal obstruction syndrome (SOS)