Testing the Addition of M6620 (VX-970, Berzosertib) to Usual Chemotherapy and Radiation for Head and Neck Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed head and neck squamous cell cancer (HNSCC) including paranasal sinus cancers but excluding nasopharyngeal carcinomas * Clinical staged III or IV HNSCC, according to American Joint Committee on Cancer (AJCC) 7th Edition, that is not amenable to surgical resection * Carcinoma of the neck of unknown primary site origin (regardless of HPV/p16 status) is eligible * Age \>= 18 years; because no dosing or adverse event data are currently available on the use of M6620 (VX-970, berzosertib) in combination with cisplatin in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 3 months * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 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 M6620 (VX-970, berzosertib) on the developing human fetus are unknown; for this reason and because DNA-damage response (DDR) inhibitors as well as other therapeutic agents used in this trial may have teratogenic potential, 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; 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 treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and for 6 months after completion of M6620 (VX-970, berzosertib) administration * Ability to understand and the willingness to sign a written informed consent document * Women of childbearing potential who are sexually active should be willing and able to use medically acceptable forms of contraception throughout the treatment phase of the trial and for up to 6 months following the last administration of study treatment; men who are sexually active must be willing and able to use medically acceptable forms of contraception throughout the treatment phase of the trial and for 6 months after completion of M6620 (VX-970, berzosertib) administration Exclusion Criteria: * Patients with nasopharyngeal carcinoma, skin squamous cell carcinoma (SCC), and salivary gland carcinomas are not eligible * Patients who are receiving adjuvant chemoradiation after surgical resection of the primary site of disease * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients who are receiving any other investigational agents * Patients on tacrolimus or any other immunosuppressants with significant interaction with cisplatin * Patient who requires live vaccine administration * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M6620 (VX-970, berzosertib) or cisplatin * Prior systemic chemotherapy for the current cancer (prior chemotherapy for a different cancer is allowed) * Prior receipt of radiotherapy that would result in overlap of the new and old radiation therapy fields * Uncontrolled intercurrent illness including, but not limited to: * Ongoing or active infection requiring intravenous antibiotics at the time of treatment initiation * Symptomatic congestive heart failure (requiring hospital stay within the last 6 months) * Myocardial infarction within the last 6 months * Unstable angina pectoris, cardiac arrhythmia * Psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because M6620 (VX-970, berzosertib) as a DNA-damage response (DDR) inhibitor may have the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with M6620 (VX-970, berzosertib), breastfeeding should be discontinued if the mother is treated with M6620 (VX-970, berzosertib); these potential risks may also apply to other agents used in this study * Human immunodeficiency virus (HIV)-positive patients with well-controlled disease, as determined by CD4 count and viral load, who are on antiretroviral therapy that does not contain a strong inducer or inhibitor of CYP3A4 are allowed on trial; HIV-positive patients on combination antiretroviral therapy with strong inducers or inhibitors of CYP3A4 are ineligible because of the potential for pharmacokinetic interactions; patients with poorly controlled HIV are not eligible due to the increased risk of lethal infections when treated with marrow-suppressive therapy * Definitive clinical or radiographic evidence of distant metastasis or adenopathy below the clavicles * M6620 (VX-970, berzosertib) is primarily metabolized by CYP3A4; therefore, concomitant administration with strong inhibitors or inducers of CYP3A4 should be avoided; because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference for a list of drugs to avoid or minimize use of; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product