A low-quality contest between the league's bottom two teams ended in stalemate as Carrick's Cathair Friel cancelled out Cahal McGinty's first-half header.
Full-back McGinty's goal was Dungannon's first for 502 minutes and ended a run of six straight defeats but it was only enough to earn their seventh point from 51 available on the road.
McGinty should have given Swifts the lead 13 minutes before he was presented with a second chance but, probably so surprised the ball reached him from Michael Carvill's corner, he met the ball with his foot six yards out and somehow hit the ball over the bar.
Rhyss Campbell forced Aaron Hogg into his first save in the 20th minute with a shot from 10 yards which the keeper gathered at his near post but, four minutes later, the Carvill-McGinty partnership ended Swifts' goal famine.
A free-kick on the left sailed over the home defence to McGinty, again unmarked but this time at the back post, and he couldn't miss from point-blank range.
Carrick were unbeaten in their last three home matches but it has hardly been home sweet home for them this season. They had only won two of 15 games at the Loughshore Hotel Arena, picking up 10 of their 19 points.
There was nothing in their first-half efforts which offered any hope they would improve that record but they had an early chance after the break when Jordan Gibson headed on a long throw but Friel's header was wastefully off target.
However, just before the hour, the striker, who is bound for Coleraine next season, made no mistake. Lloyd Anderson, with what was his last touch before being substituted, won the tackle in the box and the ball broke for Friel who placed his shot into the corner of the net.
Jordan Jenkins, who replaced Anderson, had the ball in the net on 75 minutes but his shot was ruled out for offside.
Rory Patterson appeared to have a great chance to win it for Swifts in the 89th minute but his side-foot shot, with the goal gaping, was blocked by a defender.
Carrick Rangers: Hogg; Forsythe, Rodgers, Ramsey, Neale; Cherry (McMullan 65), Kelly, Chapman, Gibson, Anderson (Jenkins 60); Ca Friel. Unused subs: Co Friel, Loughran, Smith, Glendining, Storey.
Dungannon Swifts: Carroll; McGee, C McGinty, M Smyth, Glenny; Convie, Carvill (Coyle 84), O Smyth, S McGinty (Patton 67); Patterson, Campbell. Unused subs: Byrne, Teggart, Byers, Mayse, O'Kane.
Referee: Steven Gregg
Man of match: Cathal McGinty
Match rating: 5/10.