We study a two-population model of bacterial competition that incorporates spatial movement via chemotaxis by coupling Lotka-Volterra competition dynamics with the Keller-Segel chemotaxis model into a system of three partial differential equations. We show that the spatial dynamics allow for coexistence of the populations even when the space-free system excludes this outcome. Finally, we show that by varying competitive strength and chemotactic sensitivity, three distinct outcomes are possible: competitive exclusion, temporally-stable coexistence, and oscillatory coexistence.