Clarets boss Eddie Howe was left cursing his former striker Brett Pitman whose opener just before half-time was followed by Nicky Maynard's 70th-minute strike to seal a Robins win.
It meant a third consecutive game without victory for the Clarets who have now missed valuable chances to force their way into the top-six.
Howe's side lost at home to Millwall last weekend before a mid-week draw with managerless Coventry, prompting the young manager to promise changes for his side's trip to Ashton Gate.
He stuck to his word with veteran defender Andre Bikey, Wade Elliott and Marvyn Bartley recalled in favour of Michael Duff and on-loan striker Nathan Delfouneso while Dean Marney out missed with a knee complaint.
The Robins also made three changes with Damion Stewart, Jamal Campbell-Ryce and Pitman drafted in.
Campbell-Ryce was the influential figure as the hosts made a promising start.
The winger fizzed a shot from just outside the area wide after eight minutes before his inswinging corner to the far post was headed over by Marvin Elliott.
Burnley first threatened in the 21st minute when Chris Eagles caught a 25-yard free-kick sweetly but directed it straight into the arms of David James.
Campbell-Ryce was soon making a menace of himself again and was only denied the opening goal by a finger-tip save from Lee Grant from close range.
Burnley finally enjoyed a sustained spell of pressure just after the half-hour with Ross Wallace, Chris Eagles and Elliott shooting wide.
Eagles then saw James push a 20-yard shot over but just as the visitors seemed to be getting a foothold in the match they fell behind on 43 minutes.
Pitman, who was a prolific scorer while working under Howe at Bournemouth, latched on to a right-wing cross from Albert Adomah before firing a shot from 12 yards into the top-left corner.
City just about deserved their lead at the break and should have had some breathing space soon after the resumption when Maynard fluffed a chance from six yards.
Burnley had chances to draw level with Bikey heading a Wallace free-kick over when well placed before Daniel Fox curled a free-kick into the arms of James.
But City earned themselves a two-goal advantage to soothe the nerves of the home fans when Maynard fired in a low shot from close-range with 20 minutes left on the clock.
The goal sucked the life out of Burnley's already faltering challenge as they failed to conjure up anything more to concern James in the City goal.
Instead the former England international could afford to watch on as Adomah forced a couple of saves from his opposite number Grant, while Lee Johnson fired over from 12 yards, but the Robins had already done enough.