We introduce and theoretically study the plasmon-photogalvanic effect in a planar noncentrosymmetric plasmonic crystal containing a homogeneous two-dimensional electron system gated by a periodic metal grating with an asymmetric unit cell. The plasmon-photogalvanic dc current arises due to the two-dimensional electron drag by the noncentrosymmetric plasmon modes excited under normal incidence of terahertz radiation. We show that the collective plasmon modes of the planar plasmonic crystal become strongly noncentrosymmetric in the weak-coupling regime of their anticrossing. A large plasmon wave vector (which is typically by two-three orders of magnitude greater than the terahertz photon wave vector) along with strong near-field enhancement at the plasmon resonance make the plasmonic drag a much stronger effect compared to the photon drag observed in conventional two-dimensional electron systems.