The transformative influence of compressibility on the Kelvin-Helmholtz instability (KHI) at the interface between two fluid streams of different velocities is explicated. When the velocity difference is small (subsonic), shear effects dominate the interface flow dynamics causing monotonic roll-up of vorticity and mixing between the two streams leading to the KHI. We find that at supersonic speed differentials, compressibility forces the dominance of dilatational (acoustic) rather than shear dynamics at the interface. Within this dilatational interface layer, traveling pressure waves cause the velocity perturbations to become oscillatory. We demonstrate that the oscillatory fluid motion reverses vortex roll-up and segregates the two streams leading to KHI suppression. Analysis and illustrations of the compressibility-induced suppression mechanism are presented.