The excess extreme-ultraviolet (EUV) emission detected in the Virgo and Coma clusters is explained by inverse Compton scattering of cosmic microwave background photons, which are scattered by the relativistic electrons that account for the extended radio synchrotron emission of these clusters. The lower limits of the average magnetic fields of these clusters estimated from the EUV excess are close to the equipartition magnetic fields derived from radio observations, indicating that the electron energies and magnetic field energies might be close to equipartition. The excess emission suggests energy reservoirs of similar to 10(61) and similar to 10(60) ergs for the Coma and Virgo clusters, respectively.