Research on selective attention has largely focused on the enhancement of behaviorally important information, with less focus on the suppression of distracting information. Enhancement and suppression can operate through a push−pull relationship attributable to competitive interactions among neural populations. There has been considerable debate, however, regarding (1) whether suppression can be voluntarily deployed, independent of enhancement, and (2) whether voluntary deployment of suppression is associated with neural processes occurring prior to the distractor onset. Here, we investigated the interplay between pre- and post-distractor neural processes, while male and female human subjects performed a visual search task with a cue that indicated the location of an upcoming distractor. We utilized two established EEG markers of suppression: the distractor positivity (PD) and alpha power (∼8–15 Hz). The PD—a component of event-related potentials—has been linked with successful distractor suppression, and increased alpha power has been linked with attenuated sensory processing. Cueing the location of an upcoming distractor speeded responses and led to an earlier PD, consistent with earlier suppression due to strategic use of a spatial cue. In comparison, higher predistractor alpha power contralateral to distractors led to a later PD, consistent with later suppression. Lower alpha power contralateral to distractors instead led to distractor-related attentional capture. Lateralization of alpha power was not linked to the spatial cue. This observation, combined with differences in the timing of suppression—as indexed by earlier and later PD components—demonstrates that cue-related, voluntary suppression can occur separate from alpha-related gating of sensory processing.