The photoresponse of graphene has recently received considerable attention. The main mechanisms yielding a finite dc response to an oscillating radiation field which have been investigated include responses of photovoltaic, photothermoelectric, and bolometric origin. In this article, we present a fully analytical theory of a photoresponse mechanism which is based on the excitation of plasma waves in a gated graphene sheet. By employing the theory of relativistic hydrodynamics, we demonstrate that plasma-wave photodetection is substantially influenced by the massless Dirac fermion character of carriers in graphene, and that the efficiency of photodetection can be improved with respect to that of ordinary parabolic-band electron fluids in semiconductor heterostructures.