The invention relates to a gear engagement device for a motor-vehicle gearbox, with dampening of the relative rotations of its components. The gear engagement device comprises a gearbox shaft, a gearwheel freely rotatably mounted on the shaft, a hub rigidly connected in rotation to the hub, and a gear engagement sleeve connected in rotation to the hub. The sleeve is axially movable relative to the hub from a neutral position towards a gear engagement position in which an auxiliary toothed crown carried by the gear wheel is in engagement with an inner toothed crown of the sleeve so as to connect in rotation the gear wheel to the shaft. The gear engagement device further comprises a synchronizer ring connected in rotation to the hub and cooperating with a conical portion of the gear wheel.The device is configured so that in the gear engagement position a conical surface of the synchronizer ring or a surface of the sleeve are axially pressed in friction contact against said conical portion of the gear wheel or against a respective engagement surface of the gear wheel, respectively, so that this friction contact tends to brake any relative rotations between the gear wheel and the synchronizer ring or between the gear wheel and the sleeve.