Improving teacher well-being at work is a great challenge worldwide. Understanding the stressors of Chinese university teachers in teaching activities is critical for shedding light on well-being in the midst of the rapid expansion of the higher education system and the quest to rise in world rankings. This study integrates the well-being perspective and the transactional model of stress and coping to investigate the mechanisms underlying the effect of challenge—hindrance stressors on teacher engagement. Data were collected through the online platform SoJump in mainland China (N = 7743), and structural equation modeling was used to test the relationship between challenge—hindrance stressors and teaching engagement. The statistical results revealed the following: (1) challenge stressors had a significant positive effect on teaching engagement, while hindrance stressors were negatively related to teaching engagement; (2) challenge and hindrance stressors were significant negative predictors of teacher job satisfaction; (3) teacher job satisfaction suppressed the impact of challenge stressors on teaching engagement and partially mediated the process by which hindrance stressors impact teaching engagement. The findings suggest that the theoretically opposing effects of the two stressors are not absolute and that special consideration should be given to teachers’ job satisfaction in relation to stress management for university teachers.