Strongly coupled optical microcavities containing different exciton states permit the creation of hybrid-polariton modes that can be described in terms of a linear admixture of cavity-photon and the constituent excitons. Such hybrid states have been predicted to have optical properties that are different from their constituent parts, making them a test bed for the exploration of light-matter coupling. Here, we use strong coupling in an optical microcavity to mix the electronic transitions of two J-aggregated molecular dyes and use both non-resonant photoluminescence emission and photoluminescence excitation spectroscopy to show that hybrid-polariton states act as an efficient and ultrafast energy-transfer pathway between the two exciton states. We argue that this type of structure may act as a model system to study energy-transfer processes in biological light-harvesting complexes.