Avelumab and Azacitidine in Treating Patients With Refractory or Relapsed Acute Myeloid Leukemia

Inclusion Criteria: * Patients with AML who are refractory (up to salvage 2) or relapsed (up to 2nd relapse); for patients with prior myelodysplastic syndrome (MDS) or chronic myelomonocytic leukemia (CMML) or myeloproliferative neoplasm (MPN) who transformed to AML, therapy received for MDS, CMML, or MPN is NOT considered as prior therapy for AML * Prior therapy with hydroxyurea, chemotherapy, biological or targeted therapy (e.g. FLT3 inhibitors, other kinase inhibitors), or hematopoietic growth factors is allowed * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Total bilirubin =\< 1.5 times upper limit of normal (x ULN) (=\< 3 x ULN if considered to be due to leukemic involvement or Gilbert's syndrome) * Aspartate aminotransferase or alanine aminotransferase =\< 2.5 x ULN (=\< 5.0 x ULN if considered to be due to leukemic involvement) * Estimated creatinine clearance \>= 30 mL/min according to the Cockcroft-Gault formula (or local institutional standard method) * Patients must provide written informed consent * In the absence of rapidly progressing disease, the interval from prior treatment to time of initiation of 5-azacytidine and avelumab will be at least 14 days OR at least 5 half-lives for cytotoxic/noncytotoxic agents, whichever is longer; the toxicity from prior therapy should have resolved to grade =\< 1, however alopecia and sensory neuropathy grade =\< 2 is acceptable; the half-life for the therapy in question will be based on published pharmacokinetic literature (abstracts, manuscripts, investigator brochures, or drug-administration manuals) and will be documented in the protocol eligibility document; use of hydroxyurea for patients with rapidly proliferative disease is allowed before the start of study therapy and will not require a washout; concurrent therapy for central nervous system (CNS) prophylaxis or continuation of therapy for controlled CNS disease is permitted; patients with CNS disease or leukemic brain metastasis must have been treated locally and be clinically stable for at least 2 weeks prior to enrollment and have no ongoing neurological symptoms that are related to the CNS disease (sequelae that are a consequence of the treatment of the CNS disease are acceptable) * Females must be surgically or biologically sterile or postmenopausal (amenorrheic for at least 12 months) or if of childbearing potential, must have a negative serum or urine pregnancy test within 72 hours before the start of the treatment * Women of childbearing potential must agree to use an adequate method of contraception during the study and until 3 months after the last treatment; males must be surgically or biologically sterile or agree to use an adequate method of contraception during the study until 3 months after the last treatment; adequate methods of contraception include: * Total abstinence when this is in line with the preferred and usual lifestyle of the patient; periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception * Female sterilization (have had surgical bilateral oophorectomy with or without hysterectomy) or tubal ligation at least six weeks before taking study treatment; in case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment * Male sterilization (at least 6 months prior to screening); for female patients on the study, the vasectomized male partner should be the sole partner for that patient * Combination of any of the two following (a+b or a+c or b+c) * a. Use of oral, injected or implanted hormonal methods of contraception or other forms of hormonal contraception that have comparable efficacy (failure rate \< 1%), for example hormone vaginal ring or transdermal hormone contraception * b. Placement of an intrauterine device (IUD) or intrauterine system (IUS) * c. Barrier methods of contraception: condom or occlusive cap (diaphragm or cervical/vault caps) with spermicidal foam/gel/film/cream/ vaginal suppository In case of use of oral contraception, women should have been stable on the same pill before taking study treatment * Note: oral contraceptives are allowed but should be used in conjunction with a barrier method of contraception due to unknown effect of drug-drug interaction * Women are considered post-menopausal and not of child bearing potential if they have had 12 months of natural (spontaneous) amenorrhea with an appropriate clinical profile (e.g. age appropriate, history of vasomotor symptoms) or have had surgical bilateral oophorectomy (with or without hysterectomy) or tubal ligation at least six weeks ago; in the case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment is she considered not of child bearing potential Exclusion Criteria: * Patients with a known allergy or hypersensitivity to avelumab, 5-azacytidine, or any of their components; known severe hypersensitivity reactions to monoclonal antibodies (grade \>= 3 National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\] 4.03), any history of anaphylaxis, or uncontrolled asthma (that is, 3 or more features of partially controlled asthma) * Patients with a known history of severe interstitial lung disease or severe pneumonitis or active pneumonitis/pneumonia or pulmonary pathology that is not well controlled in the opinion of the treating physician and/or principal investigator (PI) * Patients who have previously been treated with avelumab (or another PD1/PDL1 inhibitor) in combination with 5-azacytidine will be excluded * Persisting toxicity related to prior therapy of grade \> 1 NCI-CTCAE v 4.03; however, alopecia and sensory neuropathy grade =\< 2 is acceptable * Active autoimmune disease that might deteriorate when receiving an immunostimulatory agent: a) subjects with diabetes type I, vitiligo, psoriasis, hypo- or hyperthyroid disease not requiring immunosuppressive treatment are eligible; b) subjects requiring hormone replacement with corticosteroids are eligible if the steroids are administered only for the purpose of hormonal replacement and at doses =\< 10 mg or 10 mg equivalent prednisone per day; c) administration of steroids through a route known to result in a minimal systemic exposure (topical, intranasal, intro-ocular, or inhalation) are acceptable * Patients with organ allografts (such as renal transplant) are excluded * Patients who are \< 90 days post allogeneic stem cell transplant will be excluded; patients beyond 90 days post-allogeneic stem cell transplant with active uncontrolled graft versus host disease (GVHD) \> grade 1 will be excluded; patients who are on a stable dose of immunosuppressive therapy (tacrolimus, cyclosporine, or other) for \> 2 weeks will be eligible but those with recent increase in the immunosuppressive medication dose within last 2 weeks to control GVHD will not be included; Note: subjects may be using systemic corticosteroids or topical or inhaled corticosteroids post allogeneic stem cell transplant; patients requiring \>= 1 mg/kg prednisone for GVHD management at the time of screening will not be eligible until the prednisone can be weaned to \< 1 mg/kg; such patients should be monitored for at least 14 days and if no flare of GVHD requiring re-escalation of steroids or additional interventions for the GVHD they will be eligible * Patients with symptomatic CNS leukemia or patients with poorly controlled CNS leukemia * Active and uncontrolled disease/(active uncontrolled infection, uncontrolled hypertension despite adequate medical therapy, active and uncontrolled congestive heart failure New York Heart Association \[NYHA\] class III/IV, clinically significant and uncontrolled arrhythmia) as judged by the treating physician * Patients with known human immunodeficiency virus seropositivity will be excluded * Known to be positive for hepatitis B by surface antigen expression; known to have active hepatitis C infection (positive by polymerase chain reaction or on antiviral therapy for hepatitis C within the last 6 months) * Any other medical, psychological, or social condition that may interfere with study participation or compliance, or compromise patient safety in the opinion of the investigator * All other significant diseases (for example, inflammatory bowel disease, uncontrolled asthma), which, in the opinion of the investigator, might impair the subject's tolerance of trial treatment * Patients unwilling or unable to comply with the protocol * Pregnant or breastfeeding * Known alcohol or drug abuse within the last 1 year * Vaccination within 4 weeks of the first dose of avelumab and while on trial is prohibited except for administration of inactivated vaccines * Acute promyelocytic leukemia (APL) * Subject has a history of other malignancies prior to study entry, with the exception of: adequately treated in situ carcinoma of the cervix uteri or carcinoma in situ of breast; basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin; previous malignancy confirmed and surgically resected (or treated with other modalities) with curative intent or completed definitive therapy (chemotherapy, radiation, others) for the malignancy at least 1 year prior to the date of screening