The apparatus has a closed housing (4, 29), one side of the housing (4, 29) being the filling side (13), which, during filling of the container (2), is located opposite an introduction opening (6) provided in said container. The housing (4) has a filling tube (3, 33) passing through it axially in relation to the axis of the introduction opening (6), said filling tube, on the filling side (13), protruding to such an extent that, in the filling position, projects into the introduction opening (6). The interior of the housing (4, 29) is connected to an external arrangement for extracting dust by suction, and the filling side (13) has an air-inlet opening. In respect of the process, for the purpose of filling the container (2), the filling side (13) of the housing (4) is positioned above the introduction opening (6) of the container (2) such that a gap (a) is produced between the filling side (13) and the introduction opening (6), and the arrangement for extracting dust by suction is set in operation at least during filling of the container (2).