Cisplatin, Intensity-Modulated Radiation Therapy, and Pembrolizumab in Treating Patients With Stage III-IV Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * STEP 1 (REGISTRATION) * Pathologically (histologically or cytologically) proven diagnosis of head and neck squamous cell carcinoma (HNSCC) involving the oral cavity (excluding lips), oropharynx (p16 negative), hypopharynx or larynx * Patients must have undergone gross total surgical resection of high-risk oral cavity, oropharynx (p16 negative), larynx, or hypopharynx squamous cell carcinoma (SCC) within 63 days prior to registration; note: patients may have a biopsy under general anesthesia in an operating room followed by definitive ablative cancer surgery representing gross total resection; the gross total resection has to be done within 63 days prior to registration; if, however, patients have ablative resection but demonstrate rapid gross recurrence or are determined to have gross persisting disease requiring re-resection to achieve gross total resection, then the patient is not eligible * Patients must have at least one of the following high risk pathologic features: * Extracapsular nodal extension * Invasive cancer at the primary tumor resection margin (tumor on ink); Note: Patients who have a positive margin and undergo re-resection with final negative margin are eligible only if they can be enrolled within 63 days of initial gross total resection AND extracapsular nodal extension was also present; patients who have a positive margin and undergo re-resection with final negative margin and do not have extracapsular nodal extension, are NOT eligible * Pathologic stage III or IV HNSCC, including no distant metastases, based on the following minimum diagnostic workup: * General history/physical examination by a radiation oncologist and/or medical oncologist within 84 days prior to registration * Examination by an ear nose and throat (ENT) or head \& neck surgeon prior to surgery; a laryngopharyngoscopy (mirror and/or fiberoptic and/or direct procedure), if appropriate, is recommended but not required; intra-operative examination is acceptable documentation * Pre-op Imaging of the head and neck: a neck computerized tomography (CT) (with contrast) or CT/positron emission tomography (PET) (with contrast) and/or an magnetic resonance imaging (MRI) of the neck (T1 with gadolinium and T2) within 84 days prior to surgery; note: this imaging data (diagnostic pre-operative scan showing gross disease) is to be submitted in Digital Imaging and Communications in Medicine (DICOM) format via transfer of images and data (TRIAD); the report is to be uploaded into Rave * Chest imaging with either a CT scan (with or without contrast) or CT/PET (with or without contrast) that includes the chest within 120 days prior to registration; Note: if the CT/PET with or without contrast is done within 84 days prior to surgery, it fulfills the chest imaging requirement * For patients with oropharyngeal cancer only: the institution will do p16 testing, and if p16 is negative, this tissue must be submitted for central review for confirmation before Step 2 registration; note: if the institution finds that the patient is p16 positive, the patient is excluded from this trial on the basis of distinct biology, prognosis, and low- or intermediate-risk rather than high-risk status * Zubrod performance status of 0-1 within 28 days prior to registration * Absolute neutrophil count (ANC): \>= 1,500 /mm\^3 * Platelets: \>= 100,000 / mm\^3 * Hemoglobin: \>= 8.0 g/dL (note: the use of transfusion or other intervention to achieve hemoglobin \[Hgb\] \>= 8.0 g/dl is acceptable) * Creatinine clearance (CrCl) \>= 50 ml/min within 14 days prior to registration as determined by 24-hour collection or estimated by Cockcroft-Gault formula * Serum total bilirubin: =\< 1.5 X ULN OR * Direct bilirubin: =\< ULN for patients with total bilirubin levels \> 1.5 ULN * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN * International normalized ratio (INR) or prothrombin time (PT): =\< 1.5 X ULN unless patient is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants * Activated Partial Thromboplastin Time (aPTT): =\< 1.5 X ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * The following assessments are required within 14 days prior to registration: sodium (Na), potassium (K), chlorine (Cl), glucose, calcium (Ca), magnesium (Mg), and albumin; note: patients with an initial magnesium \< 0.5 mmol/L (1.2 mg/dl) may receive corrective magnesium supplementation but should continue to receive either prophylactic weekly infusion of magnesium and/or oral magnesium supplementation (e.g., magnesium oxide) at the investigator's discretion * For women of childbearing potential, a negative serum pregnancy test within 14 days of registration * Female patients of childbearing potential and men receiving MK-3475 (pembrolizumab) who are sexually active with women of childbearing potential must be willing to use an adequate method of contraception for the course of the study through 120 days after the last dose of MK-3475 (pembrolizumab); note: abstinence is acceptable if this is the usual lifestyle and preferred contraception for the patient * Patients with feeding tubes are eligible for the study * The patient or a legally authorized representative must provide study-specific informed consent prior to study entry, including consent for mandatory tumor tissue, serum, and blood submission for immune correlatives (all patients) and p16 analysis (oropharyngeal cases only) * STEP 2 (REGISTRATION) * For patients with oropharyngeal cancer only: p16 negative, confirmed by central pathology review Exclusion Criteria: * Definitive clinical or radiologic evidence of metastatic disease * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 1095 days (3 years); noninvasive cancers (for example, carcinoma in situ of the breast, oral cavity, or cervix) are permitted even if diagnosed and treated \< 3 years ago * Patients with simultaneous primaries or bilateral tumors are excluded, with the exception of patients with bilateral tonsil cancers or patients with T1-2, N0, M0 differentiated thyroid carcinoma, who are eligible * Prior systemic therapy, including cytotoxic chemotherapy, biologic/targeted therapy, or immune therapy for the study cancer; note: prior cytotoxic chemotherapy or biologic/targeted therapy for a different cancer is allowable; however, a prior anti-programmed cell death (PD)-1, anti-PD-L1, or anti-programmed cell death 1 ligand 2 (PD-L2) agent is not permitted * Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields * Severe, active co-morbidity defined as follows: * Unstable angina and/or congestive heart failure requiring hospitalization within 6 months prior to registration * Transmural myocardial infarction within 6 months prior to registration * Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration; Note: if the infection resolves and the patient is on oral (p.o.) and still within, the required registration timeframe, then the patient is eligible * Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration * Idiopathic pulmonary fibrosis or other severe interstitial lung disease that requires oxygen therapy or is thought to require oxygen therapy within 1 year prior to registration * History of (non-infectious) pneumonitis that required steroids or current pneumonitis * Acquired immune deficiency syndrome (AIDS) based upon current Center for Disease Control and Prevention (CDC) definition; note: human immunodeficiency virus (HIV) testing is not required for entry into this protocol; the need to exclude patients with AIDS from this protocol is necessary because the cisplatin and IMRT involved in this protocol may be significantly immunosuppressive; patients with known HIV, CD4 counts \>= 250/uL, and undetectable viral loads who are stable on an antiretroviral regimen may be included * A diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of the MK-3475 (pembrolizumab) * Known history of active TB (Bacillus tuberculosis) * Known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis c virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected); Note: patients who have been curatively treated for hepatitis C and have no detectable viral load are eligible * 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 * Grade 3-4 electrolyte abnormalities (CTCAE, v. 4): * Serum calcium (ionized or adjusted for albumin) \< 7 mg/dl (1.75 mmol/L) or \> 12.5 mg/dl (\> 3.1 mmol/L) despite intervention to normalize levels * Glucose \< 40 mg/dl (\< 2.2 mmol/L) or \> 250 mg/dl (\> 14mmol/L) * Magnesium \< 0.9 mg/dl (\< 0.4 mmol/L) or \> 3 mg/dl (\> 1.23 mmol/L) despite intervention to normalize levels * Potassium \< 3.0 mmol/L or \> 6 mmol/L despite intervention to normalize levels * Sodium \< 130 mmol/L or \> 155 mmol/L despite intervention to normalize levels * Patients who are pregnant, nursing, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of MK-3475 (pembrolizumab) * Hypersensitivity to MK-3475 (pembrolizumab) or any of its excipients; * Patients who 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 * Patients for whom it is not in the best interest to participate in the study, in the opinion of the treating investigator