1. A vehicular power-transmission control system comprising: an engine, which can be stopped when a vehicle comes into a halt in an idling elimination control; a drive-power transmission, which transmits a rotational driving force from said engine to wheels; a hydraulically actuated frictionally engaging element, which controls transmission of the rotational driving force in said drive-power transmission; an electrical motor, which is provided closer to said engine than to said frictionally engaging element and which assists said engine in rotational drive; and a hydraulic pump, which is connected to an output shaft of said engine and which supplies an engagement-actuating pressure to said frictionally engaging element; wherein: after said engine has stopped because of said vehicle having come into a halt in the idling elimination control, to restart said vehicle, said engine is driven to rotate the output shaft with only a small limited output torque, so as to drive said hydraulic pump; and when said frictionally engaging element receiving a hydraulic pressure from said hydraulic pump becomes ready for a transmission of power, said engine is operated to produce a torque.