(1) Background: During the past 3 years, the COVID-19 pandemic has severely affected the normal school schedule of college students, jeopardizing their mental health, sleep quality, and interpersonal relationships. However, previous studies have focused on the dimension of social support received, and few studies have measured in depth the association of support received from family on adolescents' physical and mental health. Therefore, this study explored the associations between family support received by Chinese college students during COVID-19 pandemic online classes, stress and sleep quality, and the mediating role of stress. (2) Methods: A cross-sectional study conducted at Chongqing Medical University recruited 712 college students through a university-wide incidental random sample using the Questionnaire Star platform. Statistical description and correlation analysis was conducted using SPSS 25.0, and structural equation modeling was constructed using AMOS 22.0 to test for mediating effects; (3) Results: The family support score of college students during the COVID-19 pandemic online course was 19.41 ± 4.62. Correlation analysis showed that sleep quality was negatively correlated with family support (r = −0.224, p < 0.01), positively correlated with stress (r = 0.324, p < 0.01), and family support was negatively correlated with stress (r = −0.159, p < 0.01). The results of structural equation modeling showed that stress partially mediated the relationship between family support and sleep quality among college students (indirect effect = −0.150, p < 0.01, SE = 0.013,95% CI = [−0.208, −0.064]). The model R2 was 36.4%. (4) Conclusions: Schools should consider implementing sleep education, and stress relief curriculum measures to improve the quality of students' sleep, and should focus on the role that family plays during online classes. This will help students overcome the negative emotional effects of stress in the COVID-19 pandemic and improve their learning efficiency and physical and mental health.