Control device for restarting engine of vehicle

A control device for restarting an engine of a vehicle is configured to automatically stop the engine in accordance with a stop command under a predetermined stop condition, to restart the engine in accordance with a restart command under a predetermined restart condition, and to engage a predetermined clutch of a transmission by supplying a fluid to the predetermined clutch when the engine is restarted. The control device comprises a detecting means for detecting a pressure of the fluid in a fluid passage, which is connected to the predetermined clutch when the engine is restarted. The control device further comprises a fluid supply control means for changing a fluid supply method for engaging the clutch in accordance with the pressure of the fluid.