Phase I/II Study of Bosutinib in Combination With Inotuzumab Ozogamicin in CD22-positive PC Positive ALL and CML

Inclusion Criteria: * Relapsed or refractory B-cell ALL or CML in lymphoid blast phase; Philadelphia chromosome must be present at screening (as determined by cytogenetic analysis, fluorescence in situ hybridization \[FISH\], or polymerase chain reaction \[PCR\] \[i.e., BCR-ABL positive\]); Note: patients with CML who have received treatment with tyrosine kinase inhibitors for their CML, and have progressed to lymphoid blast phase are eligible for frontline treatment; Frontline Ph+ ALL or CML-lymphoid blast phase (LBC) Cohort: Patients with newly-diagnosed Ph+ ALL or CML-LBC, who have received no or minimal treatment (minimal treatment is defined as treatment with steroids/hydroxyurea of =\< 2 week duration; vincristine =\< 2 doses; tyrosine kinase inhibitor of =\< 4 week duration; =\< 2 doses of cytarabine) and are \>= 60 years or older are eligible; patients must have bone marrow blasts \> 5% at the time of screening * Expression of CD-22 in \>= 20% blasts * Eastern Cooperative Oncology Group (ECOG) performance status score of \< or = 2 * Serum bilirubin \< or = 2.0 mg/dl * Serum creatinine \< or = 2.0 mg/dl * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \< or = 3 x upper limit of normal (ULN) * Females of childbearing potential must have a negative serum or urine beta human chorionic gonadotropin (beta-hCG) pregnancy test result within 14 days prior to the first dose of study drugs and must agree to use one of the following effective contraception methods during the study and for 30 days following the last dose of study drug; effective methods of birth control include: birth control pills, shots, implants (placed under the skin by a health care provider) or patches (placed on the skin); intrauterine devices (IUDs); condom or occlusive cap (diaphragm or cervical/vault caps) used with spermicide; females of non-childbearing potential are those who are postmenopausal greater than 1 year or who have had a bilateral tubal ligation or hysterectomy * Males who have partners of childbearing potential must agree to use an effective contraceptive method during the study and for 30 days following the last dose of study drug * Patients or their legally authorized representative must provide written informed consent Exclusion Criteria: * History of another primary invasive malignancy that has not been definitively treated or in remission for at least 2 years; patients with non-melanoma skin cancers or with carcinomas in situ are eligible regardless of the time from diagnosis (including concomitant diagnoses) * Patients with active unstable angina, concomitant clinically significant active arrhythmias, myocardial infarction within 6 months, or congestive heart failure New York Heart Association class III-IV; patients with a cardiac ejection fraction (as measured by either multi gated acquisition scan \[MUGA\] or echocardiogram) \< 40% are excluded * Known evidence of active cerebral/meningeal disease; patients may have history of central nervous system (CNS) leukemic involvement if definitively treated with prior therapy and no evidence of active disease (defined as \>= 2 consecutive spinal fluid assessments with no evidence of disease) at that time of registration * Previous treatment with any anti-CD22 directed therapy * Patients with previous allogeneic stem cell transplant (SCT) if they meet either of the following criteria: * \< 100 days from allogeneic SCT * Active acute or chronic graft-versus-host disease (GvHD), or * Receiving immunosuppressive therapy as treatment for GvHD within the last 7 days * Patients with uncontrolled active infections (viral, bacterial, or fungal) are not eligible * Active hepatitis B or C infection, or known seropositivity for human immunodeficiency virus (HIV) * Patients with liver cirrhosis or other serious active liver disease or with suspected alcohol abuse * History of autoimmune diseases (such as systemic lupus erythematosus \[SLE\], Wegener's, Wegener's granulomatosis, polyarteritis nodosa); Note: Prior autoimmune diseases are allowed as long as clinically stable * Prior chemotherapy/radiotherapy/investigational therapy within 2 weeks before the start of study drugs with the following exceptions: * To reduce the circulating lymphoblast count or palliation: steroids, hydroxyurea; no washout necessary for these agents * For ALL maintenance/CML treatment: mercaptopurine, methotrexate, vincristine, single-agent, single-dose of cytarabine and/or tyrosine kinase inhibitors; these agents should be discontinued at least 48 hours prior to start of study drugs; (Note: the interval of time from last dose of any approved tyrosine kinase inhibitor \[TKI\] to start of protocol treatment is 48 hours regardless of the indication for treatment with the TKI) * Patients who have not recovered from acute non hematologic toxicity (to =\< grade 1) of all previous therapy prior to enrollment * Females who are pregnant or lactating * Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that in the opinion of the investigator may increase the risk associated with study participation or investigational product administration or may interfere with the interpretation of study results and/or would make the patient inappropriate for enrollment into this study * Patients previously exposed to bosutinib are eligible unless they carry T315I * Patients with T315I mutations will be excluded (this criteria is not applicable for the frontline Ph+ ALL or CML-LBC cohort)