This phase I trial studies the side effects and best dose of pembrolizumab when given together with cisplatin and intensity-modulated radiation therapy, in treating patients with stage III-IV squamous cell carcinoma of the head and neck. Monoclonal antibodies, such as pembrolizumab, may block tumor growth in different ways by targeting certain cells. Drugs used in chemotherapy, such as cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Intensity-modulated radiation therapy uses high-energy x-rays to kill tumor cells and shrink tumors. Giving pembrolizumab with cisplatin and intensity-modulated radiation therapy may work better in treating patients with squamous cell carcinoma of the head and neck.