A hydraulically operated continuously variable transmission associated with the engine of a motor vehicle has a clutch for controlling the transfer of power through the transmission and a hydraulic servo unit for selectively connecting and disconnecting the clutch. A clutch control device comprises a first discharge oil passage for discharging working oil from the hydraulic servo unit when the clutch is connected by the hydraulic servo unit to start the transfer of power, an orifice disposed in the first discharge oil passage and having a prescribed cross-sectional area, a second discharge oil passage for discharging the working oil from the first discharge oil passage, the second discharge oil passage communicating with the first discharge oil passage upstream of the orifice with respect to the direction in which the working oil flows through the first discharge oil passage, and a relief valve disposed in the second discharge oil passage and openable only when the pressure of the working oil in the second discharge passage is higher than the pressure of the working oil which is developed in the first discharge oil passage upstream of the orifice when the working oil is at a normal temperature. The degree to which the flow of the working oil is restricted by the orifice may be modified by an orifice charging valve depending on the magnitude of an indication of the throttle valve opening of the engine.