Carrick moved off the bottom of the Premiership with a 1-0 victory over the Swifts on Wednesday night at Seaview.
Kyle Cherry got the winner in the 66th minute, slotting the ball past goalkeeper Stuart Addis from an Aaron Harmon cross.
The result for Carrick ended a run of five league games without a victory as they moved ahead of Warrenpoint in the relegation battle.
Dungannon remain eighth, six points clear of the relegation zone.
The Swifts have not beaten Carrick since 2011 and manager Rodney McAree will be disappointed, as his side's four-game unbeaten run came to an end.
The visitors were the more threatening team in a quiet first half, as Cormac Burke and Andrew Mitchell both went close with shots from distance.
Rangers boss Gary Haveron had stated the importance of his side's home form in their battle to stay in the top flight and they secured an important victory.
Harmon showed skill to set up Cherry and the midfielder made no mistake with the finish.
Ben Roy and Harmon went close late on to doubling Carrick's advantage but Cherry's goal was enough to seal the three points.