The unprecedented experience of national lockdowns and uncertainty of academic career due to the COVID-19 pandemic has multifaceted impacts on mental health among university students worldwide. This study determined its impact on depression and anxiety level, and associated risk factors among engineering students studying at College of Science and Technology (CST), Phuentsholing, Bhutan during the first lockdown in the country. Self-reported depression and anxiety levels were assessed using Patient Health Questionnaire (PHQ-9) and Generalized Anxiety Disorder-7 (GAD-7) respectively. Data was collected using an e-questionnaire link generated in Google form and the link was shared with students via the student’s official email group. A total of 278 students (response rate, 26.9%) completed the questionnaire. The majority of respondents were male (69.8%) and were aged from 18 to 30 (Mean: 21.7 ±SD 2.07) years. The prevalence of self-reported moderate to severe depression and anxiety were 44.2% (95% CI, 38.5–49.6) and 27.3% (95% CI, 22.3–32.4) respectively. Participants having their family members as frontline workers reported a significantly higher level of anxiety (χ2 = 4.85, p = 0.028). In multivariable logistic regression analysis, students who were academically lagging showed a higher risk of depression (AOR = 5.36, 95% CI = 2.86–10.04) and anxiety (AOR = 3.83, 95%CI = 1.86–7.88) as compared to students who were not academically behind. A high percentage of depression and anxiety was reported by students of CST during the COVID-19 pandemic. Findings from the study highlight the importance of adopting appropriate online-based teaching and learning methods to ensure timely academic and professional achievements. Moreover, the relevant stakeholders should put health system strategies in place to provide psychological support to university students during the COVID-19 pandemic.