Blair Sturrock had earlier given Swindon the lead but Vincent, who seconds before had got to his feet gingerly having been floored by a heavy tackle, collected a pass from David Bird 25 yards from goal and thundered a shot past goalkeeper Phil Smith.
It was the culmination of a gutsy fightback by the home side, who found themselves under pressure for much of the first half before manager John Ward changed the picture with a double substitution on 66 minutes.
Vincent and Damian Spencer slowly forced things Cheltenham's way, although they needed goalkeeper Shane Higgs to make a breathtaking save from Swindon substitute Miguel Comminges 15 minutes from time.
Higgs had also starred in the first half as Swindon took the game to their hosts right from the off.
They could have taken a ninth minute lead when Sturrock's header from a Christian Roberts cross was parried by Higgs. Sofiane Zaaboub, who had an excellent game for Swindon, shot from the rebound but Higgs blocked it with his legs.
Roberts and Zaaboub were causing mayhem in the opening stages and although Swindon goalkeeper Smith had to touch a long shot from Cheltenham right-back Jerry Gill over the bar after 18 minutes, it was not long before the visitors took the lead.
Zaaboub got away on the left and turned sharply, sending over a cross that Sturrock reached with a textbook downward header to give Higgs no chance.
Sturrock almost doubled the lead three minutes before half-time when he raced onto a long ball and saw his drive clawed aside by Higgs. Gill managed to scramble the ball clear with Zaaboub ready to strike.
Higgs saved again from Sturrock ten minutes into the second half but Cheltenham grew in confidence and belief as the match wore on and, thanks to Vincent's magnificent goal, made it four points to Swindon 's five from the opening three games.