Recent experiments performed on cuprates and alkali-doped fullerides have demonstrated that key signatures of superconductivity can be induced above the equilibrium critical temperature by optical modulation. These observations in disparate physical systems may indicate a general underlying mechanism. Multiple theories have been proposed, but these either consider specific features, such as competing instabilities, or focus on conventional BCS-type superconductivity. Here we show that periodic driving can enhance electron pairing in strongly correlated systems. Focusing on the strongly repulsive limit of the doped Hubbard model, we investigate in-gap, spatially inhomogeneous, on-site modulations. We demonstrate that such modulations substantially reduce electronic hopping, while simultaneously sustaining superexchange interactions and pair hopping via driving-induced virtual charge excitations. We calculate real-time dynamics for the one-dimensional case, starting from zero- and finite-temperature initial states, and we show that enhanced singlet-pair correlations emerge quickly and robustly in the out-of-equilibrium many-body state. Our results reveal a fundamental pairing mechanism that might underpin optically induced superconductivity in some strongly correlated quantum materials.