Chris Eagles attacked the Bristol defence before feeding Martin Paterson on the left and his low cross was hammered home from close range by Thompson who had replaced Joey Gudjonsson early in the second half.
Bristol's poor home record had looked set to improve when they took the lead through a 52nd-minute penalty.
Clarke Carlisle's foul on Stern John was spotted by a linesman and Nicky Maynard produced a perfect low spot-kick to find the bottom corner with his right foot.
But the goal only brought Burnley to life. Chris McCann volleyed inches wide from an Eagles cross before the equaliser arrived on 57 minutes.
An Eagles free-kick from the right was headed back across goal by Carlisle for Paterson to nod home his 13th goal of the season.
Substitute Lee Trundle twice went close to an equaliser for City, shooting narrowly wide on 62 minutes and producing a fine late save from Brian Jenson with a delicate chip.
But Burnley were always dangerous themselves and Eagles had a 67th-minute shot beaten away well by unsighted goalkeeper Adriano Basso.
Eagles became an increasing threat for the visitors as the game opened up and had good claims for man of the match.
The first half had been littered with passing errors and sloppy play.
It saw City go closest to breaking the deadlock when John lost his marker to head Liam Fontaine's 44th-minute cross wide.
Winger Michael McIndoe threatened occasionally for the home side with Burnley's Eagles doing the same for his team. Both players were narrowly off target with shots from distance.
But for most of the opening period the majority of fans in a 16,000 crowd must have wished they were Christmas shopping.
In the end there was a fair amount of excitement after Maynard's penalty and Burnley just about deserved the points.