An automatic transmission, for a vehicle with an engine, includes a hydraulic fluid pump which supplies pressurized hydraulic fluid. The engine load is detected, and the pump output is controlled so as to increase with increase of the load on the engine. If the pump is one whose output per one revolution of its input member is variable, then its output may be thus controlled by increasing its output per one revolution along with increase in engine load. And, if the pump is a variable capacity type vane type hydraulic fluid pump the output of which per one revolution of its input member is controlled by varying the eccentric position of a controlling member thereof, then its output per one revolution may be thus controlled by detecting the eccentric position of the controlling member using a sensor, and by varying this eccentric position in a feedback manner to achieve a target eccentric position which is determined to give a pump output per one revolution varying as an increasing function of engine load. Alternatively, if the pump is a fixed capacity type electrically driven type hydraulic fluid pump the output of which per one revolution of its input member is substantially constant, then its output may be thus controlled by varying the rotation speed of its input member as an increasing function of engine load. A system for implementing this method is also disclosed.