The Clarets are now without a win in 10 games in the Premier League as they failed to take their chances against Swansea City in South Wales.
Miraculously, despite that run of form, Burnley held on to seventh spot ahead of Leicester City's trip to the Etihad Stadium where they faced Manchester City.
Burnley controlled the game well in the first half and were unfortunate not to be ahead at the interval.
Alfie Mawson, the match-winner against Liverpool at the Liberty Stadium last month, impacted at the opposite end of the pitch on this occasion, blocking Jeff Hendrick's shot following Johann Berg Gudmundsson's break down the left hand side.
The Clarets went close again when Ashley Barnes directed Kevin Long's pass in to the feet of Sam Vokes; the striker did well to spin away from Federico Fernandez on the edge of the box to get a shot away but the ball travelled just past the post.
Sean Dyche's side, who had taken the lead just once in 10 games in the top flight, continued to probe for the opener and Vokes was in the thick of the action again when winning the aerial duel with Mawson to meet Charlie Taylor's cross but Barnes's attempt dropped over the crossbar.
The fixture was taking on a similar feel to Burnley's clash against Huddersfield Town at the John Smith's Stadium, with the visitors failing to fully assert their dominance, especially once Gudmundsson failed to capitalise in the next wave.
Aaron Lennon did well to engineer a crossing opportunity on the right flank, the winger's delivery was flicked on by Vokes only for the Icelandic international to slice wide of the upright.
Carlos Carvalhal's men were able to muster a shot on goal before the break but Sam Clucas fired waywardly in to the stands.
The second half was a much more even affair but it was the home side who would go on to snatch the points.
Again, the Clarets went close to beating Lukasz Fabianski but ultimately their poor goalscoring record shifted to seven in 13 games.
Barnes whipped the ball around the post early on before the Polish stopper flicked Gudmundsson's piledriver over the crossbar.
After Nick Pope dropped to his left to save Ki-Sung-Yeung's free kick, centre back Ben Mee steered a header from Gudmundsson's set-piece wide of the upright.
But City were starting to threaten and Pope somehow managed to produce a one-handed save to claw the ball off the line once Jordan Ayew got a toe to Tom Carroll's teasing delivery.
In the end, though, it was Carvalhal's changes that made the difference as the former Sheffield Wednesday boss introduced Andre Ayew and Tammy Abraham.
The pair added an extra edge to City's attack and it was the Ghanaian debutant, back for his second stint at the club, who claimed the assist for the winner.
Carroll did well to keep the ball alive and, with the play worked inside from the right, it was South Korean midfielder Ki who took a touch on the edge of the penalty area before firing in to the corner in the 81st minute.
Jordan Ayew could have added another late on when breaking free on the bye-line but he was unable to beat Pope at his near post.