Considerable research efforts are currently being devoted to analysing the role that the attentional system plays in determining driving behaviour, with the ultimate objective of reducing the number of attention-related accidents. The present study aims to assess the influence of differences in the functioning of the three attentional networks (executive control, attentional orienting and alerting) when drivers have to deal with some common hazardous situations, for example, when an oncoming car or a pedestrian unexpectedly crosses their trajectory. Multiple measures of participants’ attentional functioning were obtained from a computer-based neurocognitive test: the Attention Networks Test for Interactions and Vigilance (ANTI-V). These measures were compared to performance in a driving simulator where different types of hazardous situations were presented. Correlation and linear regression analyses revealed significant associations between individual attentional measures and driving performance in specific traffic situations. In particular, a higher attentional orienting score on the ANTI-V was associated with safer driving in situations where a single precursor anticipated the hazard source, whereas in complex situations with multiple potential hazard precursors, higher attentional orienting scores were associated with delayed braking. Additionally, partial evidence of a relationship between crash occurrence and the functioning of the executive control and the alerting networks was found. Overall, the current research would be helpful to better understand the role that each attentional network (executive control, attentional orienting and alerting) play in safe driving, and thus to develop efficient countermeasures to reduce attention-related crashes.