Saccadic eye movements are known to cause saccadic suppression, a temporary reduction in visual sensitivity and visual cortical firing rates. While saccadic suppression has been well characterized at the level of perception and single neurons, relatively little is known about the visual cortical networks governing this phenomenon. Here we examine the effects of saccadic suppression on distinct neural subpopulations within visual area V4. We find subpopulation-specific differences in the magnitude and timing of peri-saccadic modulation. Input-layer neurons show changes in firing rate and inter-neuronal correlations prior to saccade onset, and putative inhibitory interneurons in the input layer elevate their firing rate during saccades. A computational model of this circuit recapitulates our empirical observations and demonstrates that an input-layer-targeting pathway can initiate saccadic suppression by enhancing local inhibitory activity. Collectively, our results provide a mechanistic understanding of how eye movement signaling interacts with cortical circuitry to enforce visual stability.