Background: The present study aimed to explore the associations between functional capacity and global cognition, executive function and well-being in older adults. Methods: Ninety-seven older adults (age 80.6 ± 8.2 years) were examined for global cognitive function (Mini-Mental State Examination), executive function (symbol cancellation test), functional capacity (sit-to-stand tests, 6 min walk test, timed up-and-go test and handgrip strength test) and well-being (quality of life, fatigue levels, sleep quality and daily sleepiness). Adjusted partial correlations were computed to examine the associations between variables. Mediation analyses were conducted to evaluate whether functional capacity would mediate the relationships between age and cognitive or executive function. Results: Greater levels of functional capacity were associated with better performance in cognitive and executive function tests (p < 0.05). Mediation analyses revealed that functional capacity partially mediated the effects of age on global cognition and executive function (indirect effect: β = −0.11, 95% CI = −0.20 to −0.03; β = 0.34, 95% CI = 0.13 to 0.57, respectively). Increased levels of functional capacity were also associated with higher quality of life (p < 0.05, r = 0.32 to 0.41), lower fatigue levels (p < 0.05, r = 0.23 to 0.37), and better sleep quality (p < 0.05, r = 0.23 to 0.24). Conclusions: Functional capacity can mediate the effects of age on global cognition and executive function in older adults and greater levels of functional capacity are associated with improved quality of life, better sleep quality, and lower fatigue levels.