Testing the Safety and Efficacy of the Combination of the Antibody Pembrolizumab and Entinostat in Patients With Myelodysplastic Syndrome Who Are Not Responding to Hypomethylating Agents

Inclusion Criteria: * Pathologically confirmed myelodysplastic syndrome (MDS) diagnosis (regardless of initial International Prognostic Scoring System \[IPSS\] risk category) or oligoblastic acute myeloid leukemia (AML) with 21-30% bone marrow (BM) blasts in whom DNMTi have failed; patients who have developed AML after DNMTi therapy can be enrolled as long as they have initiated DNMTi therapy while they were in the MDS or oligoblastic AML (20-30% BM blasts) phase and the study chair agrees; failure of DNMTis is defined as: failure to achieve a complete response (CR), partial response (PR) or hematologic improvement (HI) after at least 4 cycles of DNMTi or progressed after such therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Calculated creatinine clearance by Modification of Diet in Renal Disease (MDRD) (CrCl) \>= 60 ml/min/1.73 squared meter * Total bilirubin =\< 2.0 mg/dL unless due to Gilbert's syndrome, hemolysis, or ineffective hematopoiesis * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x upper limit of normal (ULN) * Females of childbearing potential must have a negative serum or urine pregnancy test within 72 hours prior to start of first cycle of therapy * Patients must have no clinical evidence of central nervous system (CNS) or pulmonary leukostasis, disseminated intravascular coagulation, or CNS leukemia * Patients must have no serious or uncontrolled medical conditions * The effects of entinostat and MK-3475 (pembrolizumab) on the developing human fetus are unknown; for this reason, women of child-bearing potential and men who are sexually active with women of childbearing potential 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; 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; men who are sexually active with women of childbearing potential, treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of entinostat and MK3475 (pembrolizumab) administration * Ability to understand and the willingness to sign a written informed consent document * Patients, who relapsed 6 months after bone marrow transplant and have no evidence of active graft versus host disease and are off systemic immunosuppressant medications for at least 2 months and have received hypomethylating agents (HMA) therapy before or after transplant and meet other eligibility criteria of progression after at least 4 months of DNMTi therapy, are eligible to be enrolled in this clinical trial * Patients who are human immunodeficiency virus (HIV) positive may participate IF they meet the following eligibility requirements: * Must be on an effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * They must have a CD4 count of greater than 250 cells/mcL * They must not be receiving prophylactic therapy for an opportunistic infection Exclusion Criteria: * Any patients eligible for allogeneic stem cell transplantation (allo-SCT) and willing to undergo allo-SCT as determined at time of screening for trial; patients who are ineligible or not interested in undergoing allo-SCT will be eligible for the trial * Any serious medical condition, uncontrolled intercurrent illness (e.g., active infection, symptomatic congestive heart failure \[CHF\], unstable angina, cardiac arrhythmias, laboratory abnormalities, or psychiatric illness and/or biopsychosocial conditions that may limit compliance * Patients with known active cancers who are on therapy for those cancers at time of screening * Patients with a known positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection might be enrolled if the viral load by polymerase chain reaction (PCR) is undetectable with/without active treatment * Pregnant or breast feeding females (lactating females must agree not to breast feed while taking the study drugs) * Use of any other experimental drug or therapy within 21 days of baseline - patients who have had chemotherapy or radiotherapy within 4 weeks of entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Known hypersensitivity to MK-3475 (pembrolizumab) or history of allergic reactions to compounds of similar chemical or biologic composition to anti-PD1 or PD-L1 antibodies or entinostat * Prior treatment with any anti-PD-1 blocking therapies or histone deacetylase inhibitors (HDACi), or anti-CTLA-4 antibody, CD137 agonist or other immune activating therapy such as anti-CD 40 antibody within the last 3 months of enrollment in the study * Any history of active or severe autoimmune disease: inflammatory bowel disease, including ulcerative colitis and Crohn's disease, rheumatoid arthritis, systemic progressive scleroderma, systemic lupus erythematosus, autoimmune vasculitis (e.g., Wegener's granulomatosis), CNS or motor neuropathy considered of autoimmune origin (e.g. Guillain-Barre syndrome, myasthenia gravis, multiple sclerosis); patients with hypothyroidism with stable hormone replacement therapy dosing are allowed on study * Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis