Dungannon moved up two places to ninth in the Carling Premiership after beating Glentoran at Stangmore Park.
Johnny Topley headed against the Glens bar before Johnny Montgomery's penalty was saved by Elliott Morris after a trip on Stefan Lavery.
Topley tapped in the opener after Morris failed to hold Lavery's shot but Darren Boyce slotted in the equaliser two minutes from the break.
Darren Murphy's deflected shot won it for the Swifts on 58 minutes.
It was first victory for Rodney McAree since taking over as Dungannon boss from Dixie Robinson.