A method is provided for the control of the state of engagement of a damper clutch arranged in association with a vehicle engine, which has a fuel cut mechanism for suspending a supply of fuel during a decelerated operation, to permit connection between an input member and an output member in a torque converter arranged on an output side of the engine. Upon detection of a decelerated operation of the engine, the damper clutch is controlled to bring the input member and output member into a desired state of engagement. Here, the damper clutch is first released for a predetermined time and is then moved to an engaged side. Next, the damper clutch is feedback-controlled into a desired slip state so that the input member and the output member are allowed to rotate with a desired difference in revolution speed therebetween. A system for practicing the method is also described.