Danny Wilson's relegation strugglers got their noses in front after Billy Paynter nodded home Michael Timlin's accurate first-half free-kick before the on-loan youngster stole the hosts a share of the points in stoppage time.
The south Londoners had the best openings in a game of very few chances, with the Robins making life very difficult for Kenny Jackett's play-off contenders.
Midfielder Adam Bolder should have done better with his early header from close range that shot-stopper Petr Brezovan clung on to. The visitors grew into the game and kept possession well but their attack struggled to threaten.
Nadjim Abdou's goal-bound header from Neil Harris' brilliant cross late in the first half produced a fantastic save from Brezovan before the opening goal lit up the game at the other end.
Timlin found Paynter in space seven yards out for the youngster to power the visitors in front.
Tony Craig was lucky that referee Pat Miller missed a clash with Billy Cox on the hour mark when the Lions skipper appeared to tread on the Swindon skipper.
Brezovan was on hand to deny Harris snatching an equaliser with a curling free-kick on 75 minutes that he tipped over the bar from the very top corner.
But on-loan Reading winger Henry salvaged a late point with his belter from distance. Having picked up the ball on the left, he darted off his flank before picking out the top corner from 25 yards.
Swindon boss Wilson will feel hard done by, while the Lions keep pace with the League One leaders in the promotion race.