City led through a 43rd minute Leon Best header but sat back in the second half and were made to pay in the 89th minute when Chris Eagles chipped Keiren Westwood.
"I think before the game we would have been happy with a point at Burnley," said Chris Coleman to the media after the game. "They are going for the play-offs and had just beaten Wolves.
"In the first half we were not in any danger but we sat back for the second 45 minutes and I'm disappointed with that approach. "Sooner or later you are going to concede and in the end we buckled."