The Fermi Gamma-ray Space Telescope has detected the gamma-ray glow emanating from the giant radio lobes of the radio galaxy Centaurus A. The resolved gamma-ray image shows the lobes clearly separated from the central active source. In contrast to all other active galaxies detected so far in high-energy gamma-rays, the lobe flux constitutes a considerable portion (greater than one-half) of the total source emission. The gamma-ray emission from the lobes is interpreted as inverse Compton-scattered relic radiation from the cosmic microwave background, with additional contribution at higher energies from the infrared-to-optical extragalactic background light. These measurements provide gamma-ray constraints on the magnetic field and particle energy content in radio galaxy lobes, as well as a promising method to probe the cosmic relic photon fields.