Owain Tudor-Jones' strike in the 93rd minute hit David Raven on its way into the net.
The goal was no more than the visitors' second-half dominance deserved but meant that Carlisle are just one point above the drop zone.
Michael Bridges had given the Cumbrians the lead after just eight minutes when Evan Horwood's cross was half cleared and the striker curled home from 15 yards.
The former Premier League striker had another chance midway through the half after good work from Cleveland Taylor down the right.
The winger picked out Bridges but his weak effort went straight to Phil Smith.
Bridges was Carlisle's main attacking option and he had two further chances in the first half but could not find the net.
Those misses proved costly and after the break Danny Wilson's men came out fighting.
Carlisle keeper Ben Williams made a fantastic one-handed save just after the restart from a vicious Hal Robson-Kanu volley.
Swindon striker Billy Paynter sliced over when he should have done better.
Just after the hour mark the Robins went close again when Jack Smith's initial shot hit the post and Simon Cox somehow missed from inside the six-yard box.
Cox then shot wide as Carlisle had to absorb a lot of pressure.
Taylor had Carlisle's only chance of the second period but his back-post header went just wide.
As the final whistle loomed, Raven gave away a needless free kick 20 yards out and Tudor-Jones stepped up to level the match.
A chorus of boos erupted around Brunton Park when referee Webster brought the game to an end.