Ballymena moved above Glentoran into fifth place in the Premiership overnight with a resounding victory.
Alan Davidson fired home the opener on 27 minutes and then Alan Teggart saw his penalty saved by Niall Morgan.
Jamie Davidson poked the ball home after a goalmouth scramble after 76 and Mark Surgenor headed home unmarked on 83 to put his side in control.
Josh Cahoon cut in from the left and scored with a curling effort into the top right-hand corner for the Swifts.
Both sides came into the game with a reputation as the league's draw specialists but the visitors completely dominated the first half and David Cushley pulled his shot wide of the post in the ninth minute.
Alan Davidson shot straight at goalkeeper Morgan and then Davy Munster teed up James McCabe, but his effort flew wide.
Davidson gave his side a deserved lead with a strike into the bottom right-hand corner of the net from 12 yards out after excellent work by the busy McCabe.
Teggart was denied by Morgan from the spot after Gary Thompson had been brought down by the Swifts goalkeeper just before the break.
Jamie Davidson netted from three yards out after Dungannon failed to clear Teggart's corner and then Surgenor met Jamie Davidson's cross from the left with a nearpost header from five yards.
Stefan Lavery went close with a header for the home side and Cahoon scored on the counter-attack for a late consolation.