The winger's superb 61st-minute strike sent Turf Moor into raptures, as he drilled in from the edge of the area to settle an entertaining clash.
With victory, Burnley moved to within a point of fourth-bottom Sunderland, who were earlier thrashed 4-0 at home by Aston Villa.
For City, meanwhile, defeat leaves them five points behind leaders Chelsea but having played two games more, with Jose Mourinho's side playing one of their two matches in hand at home to Southampton on Sunday.
The champions also opened the door to those behind them chasing UEFA Champions League qualification, with Arsenal now just a point adrift in third while Manchester United, Liverpool and Tottenham will all look to take advantage of the stuttering form of Manuel Pellegrini's side.
Fernandinho, who reportedly had a bust-up with Kompany following the defeat to Liverpool, also started and he had City's first chance when he powered through midfield, but his shot was blocked and the ball hit Edin Dzeko before going out for a goal kick.
Burnley grabbed the lead against the run of play just past the hour mark when Kieran Trippier floated in a free-kick and Kompany’s headed clearance landed perfectly for Boyd on the edge of the area. The winger caught the ball with a superb left-footed half-volley, which flew past Joe Hart low into the opposite bottom corner of the goal.
A few minutes later, Michael Duff was penalised for a foul on Aguero but when the free-kick came in, City substitute Wilfried Bony’s header was easy for Heaton to deal with.
The visitors pressed for an equaliser and Fernandinho volleyed well over from another move before, with 10 minutes to go, Yaya Toure floated the ball into the Burnley area, where an unmarked Aguero headed over the crossbar.
