The visitors took an undeserved lead when a 58th-minute mistake by Lloyd Kelly allowed Sammy Ameobi to drive in a cross from the right that was headed home by Will Buckley.
But City responded to a triple substitution by head coach Lee Johnson on the hour. One of the replacements, Matty Taylor, equalised on 64 minutes with a low drive that beat Ben Alnwick at his near post.
Two minutes later the inspired Taylor latched onto a downward header from Adam Webster at a corner and volleyed the ball across goal for Palmer to nod home, the ball just crossing the line before being cleared.
The result saw the hosts move to within two points of the play-off zone, extending their unbeaten run in League and cup to 10 games, while Wanderers remain in deep trouble.
City dominated possession for much of the first half without being able to make their pressure tell.
Callum O'Dowda fired over and Josh Brownhill sent two free-kicks from central positions into the Bolton wall as Johnson's men searched for a cutting edge to some promising early approach play.
Andreas Weimann's 17th minute shot on the turn from Kelly's ball into the box threatened the Bolton goal and three minutes later Famara Diedhiou's low shot brought a save from Ben Alnwick.
Bolton began to get a foothold in the game. Craig Noone headed over on 22 minutes and moments later Sammy Ameobi strode forward powerfully to test Niki Maenpaa with a skidding drive from outside the box.
City went closest to a first-half goal on 27 minutes when Weimann drove across the edge of the penalty area only to see his low effort bounce up off a Bolton defender and onto the crossbar with Alnwick beaten.
Diedhiou fired wide after holding off a defender and at the other end Mark Beevers headed straight at Maenpaa from a Noone free-kick as the first 45 minutes ended without a breakthrough.
The second half began in similar fashion. Diedhiou headed wide at the near post and then missed a great chance to put City ahead.
Webster's header was deflected onto the bar and the ball dropped to the striker who fired badly over from six yards.
Buckley's goal came against the run of play and Johnson quickly responded by sending on Palmer, Taylor and Jay Dasilva.
The effect was immediate with Taylor claiming a goal and an assist, while Palmer produced some eye-catching runs.
Bolton applied some late pressure, but it was too little too late.