At Stangmore Park, Kobei Moore opened the scoring for the home side but Joel Cooper levelled midway through the first half before Mark Coyle sealed the points with a first goal for the Bannsiders just after half-time.
Moore opened the scoring for the hosts with less than two minutes on the clock.
Tiernan Kelly’s incisive pass split a hesitant Coleraine defence and the on-loan Larne striker raced clear of Mark Connolly and Charles Dunne before angling a low left foot drive past Ryan Schofield and into the far corner of the net.
He could have added a second minutes later but was denied by the Coleraine keeper while, at the other end, Alex Henderson held on to a long range Cooper shot at the second attempt.
Cooper wasn’t to be denied, however, and he levelled for the visitors just before the half hour.
Jamie Glackin and Lyndon Kane combined down the right hand side and when the Dungannon goalkeeper parried away Kane’s cross there was Coleraine’s leading scorer to control the ball before blasting a volley into the unguarded net for his 18th league goal of the season.
The visitors hit the front five minutes after half-time thanks to Coyle, who signed from Shelbourne in January.
The Swifts defence failed to deal with a Will Patching corner and when the ball was recycled in the direction of the Bannsiders midfielder he took a touch before rifling an unstoppable shot into the roof of the net past a helpless Henderson.
Schofield produced another important save to deny Moore just after the hour mark after the big striker had wormed his way past a number of defenders but the keeper was quickly out to smother his effort.
Cooper looked to have scored his second of the afternoon in the closing minutes but inexplicably missed the target one on one with the keeper.
