Control device for restarting engine of vehicle

A control device for restarting an engine of a vehicle configured such as to automatically stop the engine in accordance with a stop command under a predetermined stop condition, restart the engine in accordance with a restart command under a predetermined restart condition, and engage a predetermined clutch of a transmission by supplying a fluid to the predetermined clutch when the engine is restarted. The control device comprises quick pressure increase means for executing a quick pressure increase control for quickly increasing a pressure of the fluid to a control target pressure when supplying the fluid to the predetermined clutch, and rotational speed detecting means for detecting a rotational speed of a fluid pump in the transmission, wherein the quick pressure increase means determines whether the quick pressure increase control is to be executed in accordance with the rotational speed.