The Clarets handed the initiative to Birmingham City in the chase for a play-off spot as they suffered only a second league defeat of the year.
Burnley couldn’t build on a superb early strike from Rob Harker as a missed second-half penalty cost them dearly at Ashton Gate.
Jordan Cropper fired over from the spot with the chance to restore the Clarets’ lead and Bristol City struck twice more to end their losing run as they hung on after Dan Agyei did convert a penalty in the last minute.
And the dropped points mean the Clarets now trail second-placed Birmingham on goal difference with three games to go in in the Professional Development League’s northern section with the top two finishers going into the title play-offs.