Jamal Campbell-Ryce, Nicky Maynard, Jordan Spence and Jon Stead were held at bay by the remarkable Arran Lee-Barrett in the Ipswich goal in Saturday's Championship clash at Ashton Gate.
Lee Martin was dismissed for a late tackle on Kalifa Cisse before the break only for Carson to conjure up a stunning strike and snatch victory against the flow of the game.
Campbell-Ryce was first to threaten the visitors' goal, blasting from distance only for Lee-Barrett to block the shot with eight minutes played.
Ipswich could have grabbed a surprise lead in the 15th minute when Connor Wickham failed to make the most of a weighted Carlos Edwards pass, flicking the ball harmlessly against David James.
The Robins turned up the heat soon afterwards, Maynard forcing another solid save from Lee-Barrett, who was at that point unaware of the frenetic half ahead of him.
With 24 minutes played, Jamie McAllister swung his corner in to Spence at the near post but the defender could only nod wide of the post.
The Tractor Boys were reduced to 10 men four minutes later, Martin mistiming his tackle on Cisse and lunging in dangerously.
Stead was kept at arm's length shortly afterwards, conjuring up a long-range effort which Gareth McAuley cleared with ease.
The Suffolk club then faced a 10-minute barrage before the half-time break. Spence produced a dipping shot that flashed over the crossbar before Marvin Elliott's pinpoint header was plucked from the air by Lee-Barrett.
Campbell-Ryce burst into the box on 40 minutes to blast goalwards yet again, the under-siege Lee-Barrett able to deflect the shot.
Stead increased the pressure on the 27-year-old stopper with a bullet header, the Ipswich man keeping it out of the net with a convincing catch.
Albert Adomah swapped wings and crossed from the left flank, Stead and Maynard dummying shots before teeing up Campbell-Ryce whose effort was well saved by the impressive visiting keeper as Ipswich held out until the break.
Ipswich came out for the second half revitalised, youngster Carson unleashing a low drive from distance which beat James to stun the home crowd in the 52nd minute.
Stead was first to respond for the shellshocked Robins, curling in left-footed from distance but missing the target.
Grant Leadbitter was subsequently forced out of action on the hour mark after failing to recover from a heavy Damion Stewart challenge, Colin Healy his replacement.
Lee-Barrett confirmed his outstanding contribution to the victory with a last-gasp save from Brett Pitman's powerful drive as Ipswich held on.