Atomically thin layers of two-dimensional materials can be assembled in vertical stacks that are held together by relatively weak van der Waals forces, enabling coupling between monolayer crystals with incommensurate lattices and arbitrary mutual rotation1,2. Consequently, an overarching periodicity emerges in the local atomic registry of the constituent crystal structures, which is known as a moiré superlattice3. In graphene/hexagonal boron nitride structures4, the presence of a moiré superlattice can lead to the observation of electronic minibands5-7, whereas in twisted graphene bilayers its effects are enhanced by interlayer resonant conditions, resulting in a superconductor-insulator transition at magic twist angles8. Here, using semiconducting heterostructures assembled from incommensurate molybdenum diselenide (MoSe2) and tungsten disulfide (WS2) monolayers, we demonstrate that excitonic bands can hybridize, resulting in a resonant enhancement of moiré superlattice effects. MoSe2 and WS2 were chosen for the near-degeneracy of their conduction-band edges, in order to promote the hybridization of intra- and interlayer excitons. Hybridization manifests through a pronounced exciton energy shift as a periodic function of the interlayer rotation angle, which occurs as hybridized excitons are formed by holes that reside in MoSe2 binding to a twist-dependent superposition of electron states in the adjacent monolayers. For heterostructures in which the monolayer pairs are nearly aligned, resonant mixing of the electron states leads to pronounced effects of the geometrical moiré pattern of the heterostructure on the dispersion and optical spectra of the hybridized excitons. Our findings underpin strategies for band-structure engineering in semiconductor devices based on van der Waals heterostructures9.