Goodwillie, who scored the equaliser in Tuesday's 1-1 draw at home to Cove Rangers, made it 12 goals for the season with a first-half brace at Broadwood.
Craig Howie then added a third soon after half-time as Clyde claimed a convincing victory in the first meeting between the sides this season, with November's original fixture having been postponed.
The Sons, like their opponents, were playing their sixth game in 12 days but they were given a flying start when Ryan McGeever put them ahead inside three minutes.
Dumbarton midfielder Ross Forbes saw his second-half penalty saved by Jordan Hart but Adam Frizzell made it 2-0 for the hosts soon after.
East Fife managed to pull a goal back in stoppage time through Ross Dunlop but Jim Duffy's side held on to seal a first league win since December and remain a point behind Clyde.