A powertrain for a vehicle includes an engine, a transmission with an input member driven by the engine, and an output member. An electric motor is operable to drive the input member. The transmission has an electrically-actuated one-way clutch with a neutral mode in which the clutch freewheels in both directions of rotation, and a locked mode in which the clutch is locked in one direction of rotation. The transmission also has a hydraulically-actuated dual-piston clutch with a spring that biases the dual-piston clutch to an engaged state. The one-way clutch is in the neutral mode and the spring biases the dual-piston clutch to the engaged state prior to a key start of the engine. The one-way clutch is actuated to the locked state following ignition of the engine after a key start, and remains in the locked state during an autostop of the engine.