Background: Driving performance is strongly vulnerable to drowsiness and vigilance fluctuations. Excessive sleepiness may alter concentration, alertness, and reaction times. As people age, sleep undergoes some changes, becoming fragmented and less deep. However, the effects of these modifications on daily life have not been sufficiently investigated. Recently, the assessment of sleepiness became mandatory in Europe for people at risk who need the driving license release. Moreover, considering the expectation that people around the world are rapidly aging, it is necessary to investigate the relationships between senescence sleep changes, vigilance levels, and driving-related cognitive skills. Method: 80 healthy subjects (40 young adults and 40 elders) participated in the study. Sleep quality, sleepiness, and vigilance levels were assessed through the Pittsburgh Sleep Quality Index, the Karolinska Sleepiness Scale, the Epworth Sleepiness Scale, and the Psychomotor Vigilance Task (PVT). Driving-related cognitive abilities were assessed through Vienna Test System TRAFFIC, investigating selective attention, tachistoscopic perception, and risk assumption. Results: 2 × 2 between-subject ANOVAs showed less habitual sleep efficiency and worse performances in PVT in the older group. Unexpectedly, younger subjects show higher self-rated sleepiness. Moreover, older adults have lower performance in attention and perception tests, but they appear to be more cautious in situations involving traffic. Finally, the multiple regressions show age to be the only robust predictor of cognitive driving-related abilities. Conclusions: This is the first study that investigates the relationships among sleepiness/vigilance and specific driving-related cognitive skills on a sufficiently large sample. Nevertheless, the study should be considered preliminary and does not allow us to understand how specific changes in sleep architecture impact performances in the elders’ everyday life and, specifically, on driving skills.