In this Letter, a fundamentally new concept of realization of hyperbolic plasmonic metasurfaces by anisotropic gain-loss competition is proposed, and the possibility of highly directional propagation and amplification of surface plasmon polaritons is predicted. A simple realistic configuration of such a metasurface represents the periodic array of lossy metallic slabs embedded in the gain matrix. Our results may pave the way for numerous applications ranging from integrated and highly directional quantum light emitters to nonlinear-optical frequency converters.