An engine automatic stop and restart control apparatus is provided. The apparatus includes a control unit which stops and restarts an engine, an actual axle torque change speed calculation unit which calculates change speed in actual axle torque that is actually generated when the engine is restarted, an ideal axle torque change speed calculation unit which calculates change speed in ideal axle torque that corresponds to engine torque generated by the engine, a change speed difference calculation unit which calculates a change speed difference that is a difference between the change speed in actual axle torque and the change speed in ideal axle torque, and a vibration suppression control unit which executes a vibration suppression control for applying braking torque based on the change speed difference.