An enforced trip to Seaview because of a three-game sojourn there for Carrick saw four goals shared by the fellow strugglers on Thursday.
Miguel Chines opened the scoring with a first-half free-kick but that was cancelled out shortly after the interval when Terry Fitzpatrick fired home.
Conor McCloskey provided Rangers’ response with a well-taken header, but it was the Swifts who had the final say when Douglas Wilson’s volley secured a share of the spoils.
The result left them locked on 20 points, alongside Ballinamallard United who travelled to Warrenpoint on Friday, the home team maintaining their amazing resurgence with a 3-0 win and thus closed the gap to the minimum behind the trio - the relegation battle now on in earnest, and it’s a quadrangular clash with Carrick and Swifts now both firmly feeling the strain.
Carrick carved out the first chance of the match with eight minutes on the clock when McCloskey’s defence-splitting pass sent Chines scampering through on goal but stopper Andy Coleman was out quickly to block the first-time shot with his feet.
The Swifts huffed and puffed through the majority of the first 45 minutes and created little of note in front of goal.
Their only significant offering came courtesy of Cormac Burke but his neat turn on the edge of box wasn’t matched by his shot which he curled wide of the target.
With clear-cut chances at a premium, it was perhaps no surprise that Chines broke the deadlock from a set-piece just after the half-hour mark.
The Portuguese striker’s left-footed free-kick from all of 35 yards evaded everyone, including keeper Coleman and somewhat fortuitously found it’s way into the net.
It was the veteran frontman’s 11th goal of the season.
Five minutes later the ball dropped invitingly for Rangers midfielder Kyle Cherry on the edge of the 18-yard box but Coleman was equal to the midfielder’s snap-shot.
But it was Dungannon who started the brighter of the two teams after the interval and on 54 minutes Fitzpatrick brought the Swifts level with a superb 25-yard strike which flew past keeper James Taylor and in off the upright.
Carrick were back in front on 67, McCloskey meeting a cross from the left with a delightful glancing header and Coleman rooted to the spot.
Traynor could have made it 3-1 to Carrick on 74 minutes but he contrived to balloon Chines’ square ball high and wide from 15 yards.
Dungannon’s Andrew Mitchell had a penalty appeal waved away by referee Evan Boyce on 78 minutes after he tangled with Andrew Doyle inside the box.
Two minutes from time midfielder Wilson rescued a point for the County Tyrone outfit with a superb volley from 25 yards.
His delightful left-foot dipping strike nestled into the bottom corner, much to the relief of the travelling support.
Carrick Rgs: Taylor, Smyth, Traynor, Taggart, Doyle, Harmon (c), McNeill, Cherry, McCloskey, Chines, Salley. Subs: Kelly, Roy, Wright, McKeown.
Dungannon Swifts: Coleman, Montgomery (78), Hegarty, Burns, McCaffrey, Wilson, Fitzpatrick (Douglas 83), HarLeague pur, Glackin, Burke, Mitchell. Subs not used: Douglas, Mullen, Lowe.
Referee: Evan Boyce, Ballygowan