Background/Objectives: Healthcare professionals operate in complex and demanding environments characterized by high workloads, emotional strain, and organizational pressures that can undermine well-being. According to Self-Determination Theory, the fulfillment of core psychological needs (autonomy, competence, and relatedness) leads to increased job satisfaction, a key indicator of occupational well-being. Additionally, leadership plays a central role in shaping needs-fulfilling environments. Drawing on Leader–Member Exchange Theory (LMX), which emphasizes that high-quality leader-follower relationships foster greater discretion, provide learning opportunities, and build constructive team interactions, this study aimed to examine whether supportive leadership is associated with job satisfaction through the mediation of autonomy, team task cohesion, and perceived training opportunities. Methods: Data were collected from a local health authority in Northern Italy through an anonymous online survey, completed by 697 healthcare professionals, including 546 non-medical healthcare staff (primarily nurses) and 151 physicians. Structural equation modeling with a robust maximum likelihood estimator was employed to test the mediation model, including professional role as a covariate. Results: Higher LMX was positively and directly associated with job satisfaction, through the partial mediation of autonomy, team cohesion, and training opportunities, all positively associated with satisfaction. Team task cohesion showed the strongest associations with both LMX and satisfaction. Physicians reported slightly higher levels of autonomy, training opportunities, and job satisfaction than non-medical professionals. Conclusions: The findings suggest that supportive leadership contributes to healthcare professionals’ job satisfaction both directly and indirectly by contributing to core needs fulfillment. Interventions that strengthen relational quality, promote team cohesion, and enhance professional development may help sustain well-being and adaptive functioning in high-demand healthcare environments.