We investigate the escape of cosmic ray (CR) electrons from a supernova remnant (SNR) to interstellar space. We show that CR electrons escape in order, from high energies to low energies, like CR nuclei. However, the escape starts later than the beginning of the Sedov phase at an SNR age of 10(3) to 7 x 10(3) yr, and the maximum energy of runaway CR electrons is below the knee at about 0.350 TeV because, unlike CR nuclei, CR electrons lose their energy as a result of synchrotron radiation. The highest-energy CR electrons might have already been detected by the High Energy Stereoscopic System (HESS) and MAGIC as a cut-off in the CR electron spectrum, and it will be probed by the Alpha Magnetic Spectrometer (AMS-02), the Calorimeteric Electron Telescope (CALET), the Cherenkov Telescope Array (CTA) and the Large High Altitude Air Shower Observatory (LHAASO) experiments. We also calculate the spatial distribution of runaway CR electrons and their radiation spectra around SNRs. Contrary to common belief, maximum-energy photons of synchrotron radiation around 1 keV are emitted by runaway CR electrons, which have been caught up by the shock. Inverse Compton scattering by runaway CR electrons can dominate the gamma-ray emission from runaway CR nuclei via pion decay. Both are detectable by CTA and LHAASO and they can give clues to the origin of CRs and the amplification of magnetic fluctuations around the SNR. We also discuss middle-aged and/or old SNRs as unidentified very-high-energy gamma-ray sources.