We study the generation and evolution of second-order energy-density perturbations arising from primordial gravitational waves. Such "tensor-induced scalar modes" approximately evolve as standard linear matter perturbations and may leave observable signatures in the large-scale structure of the Universe. We study the imprint on the matter power spectrum of some primordial models which predict a large gravitational-wave signal at high frequencies. This novel mechanism, in principle, allows us to constrain or detect primordial gravitational waves by looking at specific features in the matter or galaxy power spectrum, thereby allowing us to probe them on a range of scales unexplored so far.