This study aimed to explore the association between sleep quality and its components and both dimensions of health-related quality of life (HRQoL) in a sample of young adults. The sample comprised 337 participants with a mean age of 19.6 y (SD = 2.22). Sleep quality and HRQoL were measured through the Pittsburgh Sleep Quality Index and the SF-12, respectively. Regression analyses were used to investigate the association between sleep quality and HRQoL. Our results confirm the significant association between sleep quality and both physical (p = 0.015; β = −0.138; R2 = 0.07) and mental (p < 0.001; β = −0.348; R2 = 0.22) HRQoL in the adjusted models. However, our results also highlight the differential association between sleep quality and mental and physical HRQoL. Whereas all the sleep quality components (except sleep latency; p = 0.349) were significantly associated with mental HRQoL (p < 0.05), just two subscales (subjective sleep quality; p = 0.021; β = −0.143 and sleep disturbances p = 0.002; β = −0.165) showed a significant association. This study showed that there is a stronger association between sleep quality and mental health than sleep quality and physical health in young adults.