Ruxolitinib in Treating Participants With Chronic Myeloid Leukemia With Minimal Residual Disease While on Therapy With Tyrosine Kinase Inhibitors

Inclusion Criteria: * Patients with Philadelphia chromosome (Ph)-positive or BCR/ABL-positive CML (as determined by cytogenetics, FISH, or PCR). * Patients must be on continuous TKI therapy for management of their CML. Any commercially available and FDA- approved TKI can be used, i.e., imatinib mesylate (IM), nilotinib (NIL) or dasatinib (DAS). Patients may be receiving TKI at entry in the frontline or salvage setting, including patients currently on imatinib after alpha-interferon failure or on dasatinib or nilotinib after failure to prior therapy including imatinib. * Patients must have received the current TKI for at least 18 months and not have increased their dose in the last 6 months. * For the phase I portion of the study, patients may be included without a CCyR provided they remain in chronic or accelerated phase CML and have at least a complete hematologic response (CHR). For the Phase II portion of the study patients must be in complete cytogenetic remission (CCyR), regardless of the stage of disease they had at the time they started therapy with TKI. * Patients must have detectable BCR-ABL transcript levels meeting at least 1 of the following criteria: Patient has never achieved a major molecular response (MMR, as defined by a BCR-ABL/ABL =\< 0.1% in the international scale (currently equivalent to 0.28 in the MD Anderson Cancer Center \[MDACC\] molecular diagnostic laboratory), and transcript levels have shown in at least 2 consecutive measures separated by at le…