Today’s COVID-19 situation can affect university Health Sciences students’ psychological health. This study aimed to analyze the stress caused by the impact of the COVID-19 pandemic on Health Sciences students from the University of Zaragoza (Spain) almost 1 year after the pandemic began. This cross-sectional descriptive study was conducted with a sample of 252 university students who completed a self-administered online questionnaire. It evaluated the impact of perceived stress with a modified scale (PSS-10-C), and assessed anxiety and depression on the Goldberg scale. Students presented stress (13.1%), anxiety (71.4%) and depression (81%). Females (81.7%) and the third-year Occupational Therapy students (p = 0.010) reported perceived stress. Nursing students perceived less stress (OR: 0.148; 95% CI: 0.026 to 0.842). University students developed stress and anxiety due to COVID-19 almost 1 year after the pandemic began. Psychological support measures for these groups should be prioritized.