The Clarets star was played in by Vincent Kompany's disastrous header to secure a vital and unexpected win for the relegation-threatened side, which is within a point of safety.
City lost another chance to catch up on leader Chelsea, while Sean Dyche's side moved to within one point of safety after a shock victory.
Vincent Kompany's wayward defensive header just after the hour proved the key moment, unwittingly setting up Boyd, who hit a thunderous first-time volley past Joe Hart to send the home crowd wild.
The visitors, though, were enraged when a stoppage-time penalty call went against them. Ben Mee appeared to catch Pablo Zabaleta rather than the ball, but the Argentine was instead punished for simulation.
The defeat leaves City five points behind leader Chelsea but having played two games more, with Jose Mourinho's side set to play one of its two matches in hand at home against Southampton on Sunday.
|VIEW FROM TURF MOOR
|By Tom Maston
The title race is over. Chelsea may as well book the open-top bus now. Manchester City has been a shadow of the Premier League winner from last season all year but, after another abject performance at Turf Moor, its race has now officially been run.
With Barcelona on the horizon, City failed to take control of the game at any point and, though it had legitimate claims for a late penalty waved away, Burnley just about deserved the three points. Certainly George Boyd's fine strike deserved to be the match-winner.
Manuel Pellegrini's side heads to Camp Nou knowing that its season could be as good as over by Wednesday night, though it must now be careful that it does not get dragged into the race for the top four. On this form, City could yet be usurped by others before the season is out.
Pellegrini recalled captain Kompany, who partnered Martin Demichelis in central defense in place of Eliaquim Mangala.
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 manager Dyche opted for leading scorer Danny Ings and Sam Vokes to lead the line but it was Boyd who threatened early on, forcing Kompany to get down well to head out his cross for a corner.
The home side saw Vokes head wide as it matched the high-flying visitors, before Dzeko fired off target at the other end.
Zabaleta typified City's growing pressure when he cut the ball back early in the second period but Aguero steered his shot past the post.
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 midfielder caught the ball with a perfect 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 penalized 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 equalizer 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.
City was upping the tempo, and in injury time it had a strong penalty claim when Mee challenged Zabaleta as he ran with the ball into a crowded area.
But referee Andre Mariner waved away the appeals, as City saw the Premier League title slip further from its grasp.