Dasatinib and Cyclosporine in Treating Patients With Chronic Myelogenous Leukemia Refractory or Intolerant to Imatinib Mesylate

Inclusion Criteria: * Patients must have histologically or cytologically confirmed chronic myelogenous leukemia (CML), Philadelphia chromosome positive (Ph+) * Patients must have a diagnosis of: * de novo chronic phase Ph+ CML, and not have received therapy with a tyrosine kinase inhibitor (TKI) for more than 2 weeks prior to enrollment * OR chronic phase Ph+ CML refractory to or with intolerance of treatment with imatinib or nilotinib therapy; patients being treated with dasatinib who have refractory disease may be considered for inclusion, at the discretion of the PIs, if other therapeutic options are not deemed likely to be efficacious (e.g. previous intolerance or refractoriness to nilotinib); evaluation for and consideration of hematopoietic stem cell transplantation as appropriate to the patient's condition by the patient's primary treating hematologist/oncologist should occur prior to enrollment in this trial * OR chronic phase Ph+ CML, without complete molecular remission after 3 months of treatment with imatinib, nilotinib or dasatinib * OR accelerated phase Ph+ CML, for which allogeneic hematopoietic stem cell transplantation is being planned, and for which no cytotoxic chemotherapy is planned prior to conditioning, and can be reasonably expected to participate for a minimum of one month prior to transplantation * OR accelerated phase Ph+ CML, for which allogeneic hematopoietic stem cell transplantation is not a therapeutic option (due to age or lack of acceptable donor, for example), and can be reasonably expected to participate for a minimum of one month * Chronic phase CML shall be defined by the presence of fewer than 15% blasts, fewer than 20% basophils, and fewer than 30% blasts plus promyelocytes in the peripheral blood and bone marrow, no extramedullary involvement except liver and spleen, and no evidence of clonal evolution (O'Brien et al., 2003) * Treatment failure/refractory disease shall be defined as less than complete hematologic response at 3 months, no cytogenetic response at 6 months, less than partial cytogenetic response at 12 months, less than complete cytogenetic response at 18 months, OR loss of CHR, loss of CCyR, clonal chromosomal abnormalities, detection of imatinib insensitive mutations, or 1 log increase in BCR-ABL transcript level from best molecular response documented on 2 samples at least one month apart (Baccarani et al., 2009) * Intolerance of TKI therapy shall be defined by non-hematologic toxic effects of any grade leading to intermittent or chronic non-compliance with, repeated dose reduction or delays in continuous dosing, or discontinuation of Imatinib * Accelerated phase CML shall be defined as the presence of \>= 15-29% blasts, \>= 20% basophils, or \>= 30% blasts plus promyelocytes in the peripheral blood or bone marrow, thrombocytopenia unrelated to therapy, or evidence of cytogenetic clonal evolution (Kantarjian et al., 1993) * Prior Therapy * Patients must have discontinued imatinib, nilotinib or dasatinib at least 7 days prior to starting study therapy; this washout period may be omitted at the discretion of the PIs, if it is determined that the washout may adversely affect patient care * Patients must discontinue hydroxyurea or interferon at least 7 days prior to starting study therapy * Life expectancy of greater than 1 month * ECOG performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,000/mcL * Absolute CD4+ count \>= 350/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * AST(SGOT)/ALT(SGPT) =\< 2.5 X institutional upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * The effects of dasatinib on the developing human fetus are unknown; for this reason and because PTK inhibitors are known to be teratogenic, women of childbearing potential must have a negative pregnancy test within 7 days of study entry; women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and for a minimum of 30 days following discontinuation of study therapy; 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; lactating women must agree not to nurse a child while on this trial or within 30 days of discontinuation of study therapy * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or failure to recover from adverse events (except alopecia) to Grade =\< 1 or to baseline (if there is persistent, chronic, stable Grade 2), due to agents administered more than 4 weeks earlier * Patients may not be receiving any other investigational agents * Known brain metastases exclude patients from this clinical trial because such patients have a poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; in addition, patients who have active brain metastases may benefit from other concurrent therapy such as radiation or radiosurgery, and should be considered for the most appropriate clinical therapy that may provide symptom relief * History of allergic reactions attributed to compounds of similar chemical or biologic composition to dasatinib or cyclosporine * Patients who require concurrent treatment with any medications or substances that are potent inhibitors or inducers of CYP3A4 are ineligible; efforts should be made to switch patients with a seizure disorder who are taking enzyme-inducing anticonvulsant agents to other medications * Patients who require concurrent treatment with proarrhythmic potential * QTc prolongation (defined as a QTc interval \>= 480 msec) or other significant ECG abnormalities * Use of antithrombotic and/or anti-platelet agents (e.g., warfarin, heparin, low molecular weight heparin, aspirin, and/or ibuprofen); exception: patients with CML who have significantly elevated platelet counts taking anagrelide are eligible; patients who require \< 2 mg of warfarin per day for central venous catheter prophylaxis are allowed on this study * Patients with any condition (e.g., gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for IV alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease) that impairs their ability to swallow and retain dasatinib tablets are excluded; tablets may not be crushed prior to administration * Patients may not have any clinically significant cardiovascular disease, defined as NYHA class III or higher and as follows: * Myocardial infarction or ventricular tachyarrhythmia within 6 months * Prolonged QTc \>= 480 msec (Fridericia correction) * Ejection fraction less than institutional normal * Major conduction abnormality (unless a cardiac pacemaker is present) * Patients with any cardiopulmonary symptoms of unknown cause (e.g. shortness of breath, chest pain, etc.) should be evaluated by a baseline echocardiogram with or without stress test as needed in addition to electrocardiogram (EKG) to rule out QTc prolongation; the patient may be referred to a cardiologist at the discretion of the principal investigator; patients with underlying cardiopulmonary dysfunction should be excluded from the study * Uncontrolled intercurrent illness including, but not limited to, the following: ongoing or active infection; history of significant bleeding disorder, including congenital (von Willebrand's disease) or acquired (anti-factor VIII antibodies) disorders; large pleural effusions; or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because animal studies with dasatinib have shown embryolethality and fetal skeletal alterations at non-toxic maternal doses; because there is an unknown but potential risk for adverse events in nursing human infants secondary to treatment of the mother with dasatinib, breastfeeding should be discontinued if the mother is treated with dasatinib * HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with dasatinib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated * Known congenital or acquired immunodeficiency * CD4+ count less than 350/µl * Mutation in Bcr-Abl known to confer resistance to dasatinib; (N.B. patients for whom Bcr-Abl mutations have not been assessed, will have this assessment in screening; they will be allowed to enroll and initiate therapy; if screening analysis reveals Bcr-Abl mutation known to confer resistance to dasatinib, the patient will be discontinued from study participation) * Prior hematopoietic stem cell transplantation