In the antisaccade task, which is considered a sensitive assay of cognitive function, a salient visual cue appears and the participant must look away from it. This requires sensory, motor-planning, and cognitive neural mechanisms, but what are their unique contributions to performance, and when exactly are they engaged? Here, by manipulating task urgency, we generate a psychophysical curve that tracks the evolution of the saccadic choice process with millisecond precision, and resolve the distinct contributions of reflexive (exogenous) and voluntary (endogenous) perceptual mechanisms to antisaccade performance over time. Both progress extremely rapidly, the former driving the eyes toward the cue early on (∼100 ms after cue onset) and the latter directing them away from the cue ∼40 ms later. The behavioral and modeling results provide a detailed, dynamical characterization of attentional and oculomotor capture that is not only qualitatively consistent across participants, but also indicative of their individual perceptual capacities.