We report a very efficient process for accelerating high-energy electrons by coherent whistler mode waves in the Earth's dipole magnetic field, which we have found in our recent test particle simulations. The efficient acceleration process takes place for weakly relativistic seed electrons of a few hundred kiloelectronvolts. Under an assumption that the whistler mode wave packets are excited near the equatorial plane of the inner magnetosphere and propagate away from the equator, the acceleration process becomes irreversible. With a sufficiently long whistler mode wave packet of the order of 1 s, the energetic electrons are accelerated to a relativistic energy range of a few megaelectronvolts through a single resonant trapping process. We call this particular acceleration process relativistic turning acceleration (RTA), which could be a viable mechanism for increasing relativistic electron fluxes in the outer radiation belt. Necessary conditions for RTA are a relatively large amplitude of whistler mode waves, in the range of 50 to a few hundred picoteslas, and an initial kinetic energy of trapped electrons in the energy range of a few hundred kiloelectronvolts. The minimum energy of electrons accelerated by the RTA process and the maximum energy attained by it are derived analytically and verified by the test particle simulations.