We suggest that the luminous extreme-ultraviolet (EUV) emission that has been detected recently from clusters of galaxies is inverse Compton scattering of cosmic microwave background radiation by low-energy cosmic-ray electrons in the intracluster medium. The cosmic-ray electrons would have Lorentz factors of gamma similar to 300 and would lose energy primarily by emitting EUV radiation. These particles have lifetimes comparable to the Hubble time; thus, the electrons might represent a relic population of cosmic rays produced by nonthermal activity over the history of the cluster. The inverse Compton model naturally explains the observed increase in the ratio of EUV to X-ray emission with radius in clusters. The required energy in cosmic-ray electrons is typically 1%-10% of the thermal energy content of the intracluster gas. We suggest that the cosmic-ray electrons might have been produced by supernovae in galaxies, by radio galaxies, or by particle acceleration in intracluster shocks.