Durvalumab Before Surgery in Treating Patients With Oral Cavity or Oropharynx Cancer

Inclusion Criteria: * Histologically or cytologically confirmed HNSCC of the oral cavity (OC; more than 90% patients have HPV negative cancer) or oropharynx (about 60-80% of patient have HPV positive cancer) * Presence of radiologically of clinically documented disease. All radiology studies must be performed within 28 days prior to registration * Any stage, considered candidates for surgery and planned for surgery either by robotic or by standard surgical technique * Documentation of HPV tested by polymerase chain reaction (PCR) (resulted or pending) * Willing to provide consent for an additional tissue biopsy for research purposes, to allow a part of their surgical tumor tissue to be utilized for research (in case tumor tissue has not already been saved in the tumor tissue bank), and to donate samples of blood and saliva collected weekly through the treatment * All patients must have provided informed consent for correlative studies * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 or 2 * Patients must have no prior exposure to immune-mediated therapy, including anti- cytotoxic T-lymphocyte protein 4 (CTLA-4), anti-programmed cell death 1, anti-programmed cell death 1 ligand 1 (PD-L1), or anti-programmed cell death ligand 2 antibodies, excluding therapeutic anticancer vaccines * At least 1 lesion, not previously irradiated, that can be accurately measured at baseline as \> or = 10 mm in the longest diameter (except lymph nodes, which must have a short axis \> or = 15 mm) with CT or magnetic resonance imaging (MRI) or clinical measurement and that is suitable for accurate repeated measurements as per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 guidelines * Previous surgery is permitted provided that a minimum of 28 days (4 weeks) have elapsed between any major surgery and date of registration, and that wound healing has occurred * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Platelet count \>= 100 x 10\^9/L * Hemoglobin \>= 9.0 g/dL * Serum bilirubin =\< 1.5 x upper limit of normal (ULN) (institutional upper limit of normal) * Total bilirubin is less than or equal to ULN, except the case in which the elevated total bilirubin is not a sign of liver disease, such as the Gilbert Syndrome, in which case a Total Bilirubin less than or equal to 2X ULN is acceptable. * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN * Serum creatinine clearance (CL) \> 40 mL/min by the Cockcroft-Gault formula or by 24-hour urine collection for determination of creatinine clearance * Female subjects must either be of non-reproductive potential (ie, post-menopausal by history: \>= 60 years old and no menses for \>= 1 year without an alternative medical cause; OR history of hysterectomy, OR history of bilateral tubal ligation, OR history of bilateral oophorectomy) or must have a negative serum pregnancy test upon study entry * In accordance with National Cancer Institute of Canada Clinical Trials Group (NCIC CTG) policy, protocol treatment is to begin within 2 working days of patient registration * Written informed consent and any locally-required authorization (e.g., Health Insurance Portability and Accountability Act \[HIPAA\] in the United States of American \[USA\], European Union \[EU\] Data Privacy Directive in the EU) obtained from the subject prior to performing any protocol-related procedures, including screening evaluations * Age 18 years or older at time of study entry. * Subject is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up Exclusion Criteria: * Involvement in the planning and/or conduct of the study (applies to both AstraZeneca staff and/or staff at the study site). Previous enrollment in the present study * Participation in another clinical study with an investigational product during the last 6 months (mo) * Any previous treatment with a PD1 or PD-L1 inhibitor, including durvalumab * Receipt of anti-cancer therapy (chemotherapy, immunotherapy, endocrine therapy, targeted therapy, biologic therapy, tumor embolization, monoclonal antibodies, other investigational agent) within the last 6 mo (before the first dose of Durvalumab). * Mean QT interval corrected for heart rate (corrected QT \[QTc\]) \>= 470 ms calculated from 3 electrocardiograms (ECGs) using Fridericia's correction * Current or prior use of immunosuppressive medication within 28 days before the first dose of durvalumab, with the exceptions of intranasal and inhaled corticosteroids or systemic corticosteroids at physiological doses, which are not to exceed 10 mg/day of prednisone, or an equivalent corticosteroid * Any unresolved toxicity (\> Common Terminology Criteria for Adverse Events \[CTCAE\] grade 2) from previous anti-cancer therapy; subjects with irreversible toxicity that is not reasonably expected to be exacerbated by the investigational product may be included (e.g., hearing loss, peripherally neuropathy) * Any prior grade \>= 3 immune-related adverse event (irAE) while receiving any previous immunotherapy agent, or any unresolved irAE \> grade 1 * Active or prior documented autoimmune disease within the past 2 years; NOTE: Subjects with vitiligo, Grave's disease, or psoriasis not requiring systemic treatment (within the past 2 years) are not excluded * Active or prior documented inflammatory bowel disease (e.g., Crohn's disease, ulcerative colitis) * History of primary immunodeficiency * History of allogeneic organ transplant * History of hypersensitivity to durvalumab or any excipient * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, uncontrolled hypertension, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, active bleeding diatheses including any subject known to have evidence of acute or chronic hepatitis B, hepatitis C or human immunodeficiency virus (HIV), or psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent * Known history of previous clinical diagnosis of tuberculosis * Receipt of live attenuated vaccination within 30 days prior to study entry or within 30 days of receiving durvalumab * Female subjects who are pregnant, breast-feeding or male or female patients of reproductive potential who are not employing an effective method of birth control * Patients with body weight \<= 30 kg * Any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results