Background: The COVID-19 pandemic and associated restrictive measures have substantially affected educational processes around the globe, resulting in psychological distress among students. The mental health of students in higher education is of paramount importance, and the COVID-19 pandemic has brought this vulnerable population into renewed focus. In this context, the evaluation of students‘ mental health at educational institutes has gained invaluable popularity during the COVID-19 pandemic. This study aimed to ascertain the psychological health and coping strategies among students from a higher education institute in Saudi Arabia. Methods: An online study instrument was used to assess anxiety (Generalized Anxiety Disorder-7, GAD-7), depression (Patient Health Questionnaire-9, PHQ-9), post-traumatic stress disorder—PTSD (Impact of Event Scale-Revised, IES-R) and coping strategies (Brief-COPE). The severity of the psychological distress was classified as per the scoring criteria and correlated with demographics using appropriate statistical methods. Results: Of 1074 students (age 21.1 ± 2.1 years), 12.9% and 9.7% had severe anxiety and depression, respectively. The mean anxiety and depression scores were 7.50 ± 5.51 and 9.31 ± 6.72, respectively. About one-third (32%) of students reported suicidal ideation, with 8.4% students having such thoughts nearly every day. The average PTSD score was 21.64 ± 17.63, where avoidance scored higher (8.10 ± 6.94) than intrusion and hyperarousal. There was no association of anxiety, depression and PTSD score with the demographics of the study participants. Religious/spiritual coping (5.43 ± 2.15) was the most adoptive coping mechanism, followed by acceptance (5.15 ± 2.10). Male students were significantly (p < 0.05) associated with active copings, instrumental support, planning, humor, acceptance and religious coping. Substance use was the least adopted coping strategy but practiced by a considerable number of students. Conclusions: The long-lasting pandemic situation, onerous protective measures and uncertainties in educational procedures have resulted in a high prevalence of psychological ailments among university students, as indicated in this study. These findings accentuate the urgent need for telepsychiatry and appropriate population-specific mental health services to assess the extent of psychological impairment and to leverage positive coping behaviors among students.