The Clarets remain nine points clear of the Premier League's drop zone despite suffering a late defeat at the hands of Swansea City at the Liberty Stadium.
Andre Gray's double had given the visitors the advantage on the hour but Paul Clement's men hit back to dramatically edge victory in a five-goal thriller in South Wales.
The Clarets rode their luck on more than one occasion in the first half and were fortunate to go in to the break on level terms.
The woodwork came to Burnley's rescue in the 11th minute when Alfie Mawson guided Gylfi Sigurdsson's set-piece against the bar and seconds later, with the ball recycled back to the defender in the box, the former Barnsley man hit the upright.
The home side, though, didn't have to wait long to open the scoring. Leroy Fer, who netted the winner at Turf Moor on the first day of the campaign, beat Jeff Hendrick down the right hand side and his delivery was converted by Fernando Llorente.
The Spaniard towered above Matt Lowton to head back across Paul Robinson and pick out the bottom corner.
City's early dominance suggested that Burnley's away day hoodoo looked set to be extended, with just two points taken from 12 games, but they responded immediately.
Hendrick tested Lukasz Fabianski with a strike from the edge of the box before the Polish international, formerly of Arsenal, thwarted Gray with a sublime one-handed save as the striker's effort flicked up awkwardly off the boot of Federico Fernandez.
Sean Dyche's side leveled from the resulting corner but their equaliser was steeped in controversy.
Record signing Robbie Brady dropped his kick in to the six yard box and, though replays highlighted that it was Sam Vokes who handled, referee Anthony Taylor stopped play and pointed to the spot.
Leading scorer Gray placed the ball down and, just like he did at the Emirates, fired the penalty down the middle in the 19th minute.
The Swans had the better of the remainder of the half and went close to restoring their lead on a number of occasions.
Robinson, replacing Tom Heaton who missed the fixture through illness, denied Martin Olsson's powerful drive with a fine save and then came out on top again when the Swede came inside on his right foot to get a shot at goal.
Sigurdsson was next, allowing the ball to roll across his body before pulling a half-volley narrowly wide as Robinson scrambled to his upright.
With the half petering out, the visitors were once again saved by the woodwork. Llorente's header from Olsson's cross looped over Robinson only to come down off the crossbar.
Then, Taylor ready to sound his whistle, Sigurdsson stung the palms of Robinson with a free kick.
City took command in the second half and looked to have added a second when Sigurdsson passed the ball alongside Robinson. However, with the ball creeping toward the corner, Ben Mee covered superbly to clear the ball off the line.
Tom Carroll grew to become a prevalent and efficient outlet for the hosts after the break and pinpoint crosses in for Llorente and Sigurdsson both resulted in his team-mates flicking efforts just past the far post.
The Clarets have only led for six minutes on the road this season, when Ashley Barnes scored against Spurs at White Hart Lane, but they added to that on the hour.
Vokes rose above his marker to flick Lowton's diagonal pass on to Gray and the 25-year-old turned instinctively before lashing the ball left-footed beyond Fabianski and in to the corner.
Again, though, Burnley failed to protect their lead. George Boyd was half asleep when failing to track Olsson's run from Sigurdsson's clever flick and the former Rovers defender broke in to the box before hammering the ball in to the roof of the net.
Olsson almost turned the game on its head moments later when his volley from the edge of the penalty area dropped just past the post while Mee looked to have secured a point for the away side when turning Carroll's shot off the line.
However, two minutes in to time added on, City snatched maximum points. Carroll was once again responsible for the assist, sitting the ball up to the back post, and Llorente over-powered Mee to win the header and beat Robinson.
Olsson went close to adding to Burnley's woes late on but Robinson showed a strong hand to beat his stinging strike to safety.