Current evidence suggests that the ability to detect and react to information under lowered alertness conditions might be more impaired on the left than the right side of space. This evidence derives mainly from right-handers being assessed in computer and paper-and-pencil spatial attention tasks. However, there are suggestions that left-handers might show impairments on the opposite (right) side compared to right-handers with lowered alertness, and it is unclear whether the impairments observed in the computer tasks have any real-world implications for activities such as driving. The current study investigated the alertness and spatial attention relationship under simulated monotonous driving in left- and right-handers. Twenty left-handed and 22 right-handed participants (15 males, mean age = 23.6 years, SD = 5.0 years) were assessed on a simulated driving task (lasting approximately 60 min) to induce a time-on-task effect. The driving task involved responding to stimuli appearing at six different horizontal locations on the screen, whilst driving in a 50 km/h zone. Decreases in alertness and driving performance were evident with time-on-task in both handedness groups. We found handedness impacts reacting to lateral stimuli differently with time-on-task: right-handers reacted slower to the leftmost stimuli, while left-handers showed the opposite pattern (although not statistically significant) in the second compared to first half of the drive. Our findings support suggestions that handedness modulates the spatial attention and alertness interactions. The interactions were observed in a simulated driving task which calls for further research to understand the safety implications of these interactions for activities such as driving.