Accurate perception of binaural cues is essential for left-right sound localization. Much literature focuses on threshold measures of perceptual acuity and accuracy. This study focused on supra-threshold perception using an anticipatory eye movement (AEM) paradigm designed to capture subtle aspects of perception that might not emerge in behavioral-motor responses, such as the accumulation of certainty, and rapid revisions in decision-making. Participants heard interaural timing differences (ITDs) or interaural level differences in correlated or uncorrelated narrowband noises, respectively. A cartoon ball moved behind an occluder and then emerged from the left or right side, consistent with the binaural cue. Participants anticipated the correct answer (before it appeared) by looking where the ball would emerge. Results showed quicker and more steadfast gaze fixations for stimuli with larger cue magnitudes. More difficult stimuli elicited a wider distribution of saccade times and greater number of corrective saccades before final judgment, implying perceptual uncertainty or competition. Cue levels above threshold elicited some wrong-way saccades that were quickly corrected. Saccades to ITDs were earlier and more reliable for low-frequency noises. The AEM paradigm reveals the time course of uncertainty and changes in perceptual decision-making for supra-threshold binaural stimuli even when behavioral responses are consistently correct.