We present direct evidence of enhanced non-radiative energy transfer between two J-aggregated cyanine dyes strongly coupled to the vacuum field of a cavity. Excitation spectroscopy and femtosecond pump-probe measurements show that the energy transfer is highly efficient when both the donor and acceptor form light-matter hybrid states with the vacuum field. The rate of energy transfer is increased by a factor of seven under those conditions as compared to the normal situation outside the cavity, with a corresponding effect on the energy transfer efficiency. The delocalized hybrid states connect the donor and acceptor molecules and clearly play the role of a bridge to enhance the rate of energy transfer. This finding has fundamental implications for coherent energy transport and light-energy harvesting.