A control device for a vehicle includes a low-pressure fuel sensor that detects a pressure of fuel stored in a low-pressure delivery pipe, and a control device that controls a feed pump based on a detected value from the low-pressure fuel sensor. The control device causes a relief valve to be opened when variation in power required in an engine continues to remain within a prescribed range for a prescribed period. With this structure, a control device for a vehicle can be provided which allows the relief valve to be opened with reduced variation in air-fuel ratio.