Background/Objectives: The burnout syndrome in nurses has been related to the development of mental health problems. On the contrary, resilience is related to adequately coping with stressful situations and better mental health. The objective was to analyze the relationship between resilience and mental health problems in nurses and estimate the proportion mediated by burnout in the association. Methods: In 2021, a total of 1165 Spanish nurses were selected through a stratified random sampling method. Participants anonymously filled in the Resilience Scale (RS-14), the Maslach Burnout Inventory Survey, and the General Health Questionnaire. To test the hypothesis proposed and explain the mediating effect of burnout empirically, structural equation modeling (SEM) was applied. A latent mediation model was computed. Results: Resilience was negatively related to burnout and mental health problems. The direct relationship between burnout and the latent health variable was positive. In addition, in view of the total effect of resilience on mental health problems and the magnitude of the indirect effect, we stated that the proportion of this effect mediated by burnout ranged from 0.486 to 0.870. Conclusions: This study reveals that fostering resilience in nurses directly and indirectly reduces burnout and improves their mental health. The implementation of resilience programs and supportive institutional policies is recommended to improve working conditions and the quality of patient care.