Pembrolizumab and Dasatinib, Imatinib Mesylate, or Nilotinib in Treating Patients With Chronic Myeloid Leukemia and Persistently Detectable Minimal Residual Disease

Inclusion Criteria: * PREREGISTRATION (STEP 0): Patient has pathologically-confirmed chronic phase-CML on a first line TKI and must meet the following criteria: * The patient has to be on first-line TKI therapy (the same TKI) for at least 2 years prior to pre-registration * Has been in MMR (i.e. MR\^3) but still have detectable BCR-ABL transcript by a standard real-time quantitative polymerase chain reaction (RQ-PCR) assay with a limit of detection (sensitivity) of 4.5 for at least 12 months from the first documentation of the MMR * Patient has not achieved MR\^4.5 (complete molecular remission \[CMR\]) within the time of initiation of TKI therapy and pre-registration * PREREGISTRATION (STEP 0): Patient must be scheduled to undergo a standard of care bone marrow biopsy within 7 days of step 0 registration * PREREGISTRATION (STEP 0): Peripheral blood must be collected for submission to Fred Hutchinson Cancer Research Center for central assessment of the establishment of BCR/ABL status to confirm patient?s eligibility for registration to Step 1; Fred Hutchinson will forward results within 1-2 business days of receipt of the peripheral blood to the submitting institution * REGISTRATION TO TREATMENT (STEP 1): Institution has received central BCR-ABL test results confirming MRD positive status * REGISTRATION TO TREATMENT (STEP 1): Patients have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * REGISTRATION TO TREATMENT (STEP 1): No active hemolytic anemia requiring immunosuppressive therapy or other pharmacologic treatment; patients who have a positive Coombs test but no evidence of hemolysis are NOT excluded from participation * REGISTRATION TO TREATMENT (STEP 1): No current use of corticosteroids; EXCEPTION: Low doses of steroids (\< 10 mg of prednisone or equivalent dose of other steroid) used for treatment of non-hematologic medical condition (e.g. chronic adrenal insufficiency) is permitted * REGISTRATION TO TREATMENT (STEP 1): No other active primary malignancy (other than non-melanomatous skin cancer or carcinoma in situ of the cervix) requiring treatment or limiting expected survival to =\< 2 years * NOTE: If there is a history of prior malignancy, they must not be receiving other specific treatment (other than hormonal therapy for their cancer) * REGISTRATION TO TREATMENT (STEP 1): Women must not be pregnant or breastfeeding; patients must also not expect to conceive or father children from the time of registration, while on study treatment, and continue for 120 days after the last dose of study treatment; all females of childbearing potential must have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of pembrolizumab; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required; a female of childbearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * REGISTRATION TO TREATMENT (STEP 1): Women of childbearing potential and sexually active males must use an accepted and effective method of contraception or to abstain from sexual from time of registration, while on study treatment, and continue for 120 days after the last dose of study treatment * REGISTRATION TO TREATMENT (STEP 1): Patients must have been on a stable dose of the TKI for the last 3 months prior to pre-registration * REGISTRATION TO TREATMENT (STEP 1): Patient may not be currently participating and receiving study therapy or have participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment * REGISTRATION TO TREATMENT (STEP 1): Patient must not have a diagnosis of immunodeficiency or be receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of treatment * REGISTRATION TO TREATMENT (STEP 1): Patient must not have a known history of active TB (Bacillus Tuberculosis) * REGISTRATION TO TREATMENT (STEP 1): Patient must not have a history of hypersensitivity to pembrolizumab or any of its excipients * REGISTRATION TO TREATMENT (STEP 1): Patient must not have received a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study registration or have not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier * REGISTRATION TO TREATMENT (STEP 1): Patient must not have had prior chemotherapy, targeted small molecule therapy (aside from imatinib, dasatinib, or nilotinib), or radiation therapy within 2 weeks prior to study registration; patients also must have recovered from all adverse events due to a previously administered agent * Note: Patients with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study * REGISTRATION TO TREATMENT (STEP 1): Patients who have received major surgery must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy * REGISTRATION TO TREATMENT (STEP 1): Patient must not have a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer * REGISTRATION TO TREATMENT (STEP 1): Patient must not have known active central nervous system (CNS) metastases and/or carcinomatous meningitis; subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of protocol treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to protocol treatment; this exception does not include carcinomatous meningitis which is excluded regardless of clinical stability * REGISTRATION TO TREATMENT (STEP 1): Patient must not have active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs); replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * REGISTRATION TO TREATMENT (STEP 1): Patient must not have known history of, or any evidence of active, non-infectious pneumonitis * REGISTRATION TO TREATMENT (STEP 1): Patient must not have an active infection requiring systemic therapy * REGISTRATION TO TREATMENT (STEP 1): Patient must not have a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator * REGISTRATION TO TREATMENT (STEP 1): Patient must not have known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * REGISTRATION TO TREATMENT (STEP 1): Patient must not have received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent * REGISTRATION TO TREATMENT (STEP 1): Patients who are Human Immunodeficiency Virus (HIV) positive are eligible if they have undetectable HIV viral load and CD4+ T-cell count ≥ 250/mm3. * REGISTRATION TO TREATMENT (STEP 1): Patients with a known positive test for Hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection might be enrolled if the viral load by PCR is undetectable with/without active treatment. * REGISTRATION TO TREATMENT (STEP 1): Patients must not have known history of hepatitis B (defined as Hepatitis B surface antigen \[HBsAg\] reactive). * REGISTRATION TO TREATMENT (STEP 1): Patient must not have received a live vaccine within 30 days of planned start of study therapy * NOTE: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed * REGISTRATION TO TREATMENT (STEP 1): Absolute neutrophil count (ANC) \>= 1,500 /mcL, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Platelet count \>= 100,000 /mcL, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Hemoglobin (Hgb) \>= 9 g/dL OR \>= 5.6 mmol/L without transfusion of erythropoietin (EPO) dependency, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR creatinine clearance (per institutional standards) \>= 60 mL/min for patient with creatinine levels \> 1.5 X ULN, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 X ULN, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for subjects with liver metastases, within 14 days prior to first dose of pembrolizumab * REGISTRATION TO TREATMENT (STEP 1): Patients should not be receiving concomitant strong CYP3A4 inducers or inhibitors ≤ 7 days prior to registration due to their potential to effect the activity or pharmacokinetics of study agents and/or QT interval prolongation toxicity. Should treatment with any of these agents be required, consult with study chair. * REGISTRATION TO TREATMENT (STEP 1): Patients should not have received prior allogeneic transplant.