Boyd scored with a brilliant low strike from the edge of the area after 61 minutes to settle a tight Barclays Premier League contest that had belatedly come to life after a drab first half.
City wanted a penalty in injury time when Pablo Zabaleta went down under a Ben Mee challenge but referee Andre Marriner gave a free-kick the other way.
The result leaves champions City still trailing Chelsea by five points at the top, but now having played two games more.
It will also increase the heat on manager Manuel Pellegrini, whose side have fallen away badly since pulling level with Chelsea on New Year's Day.
City's hopes of rescuing something from their season may now come down to the sizeable task of overturning a 2-1 deficit against Barcelona in the Champions League on Wednesday.
George Boyd scored the only goal with a superb second-half strike and the Clarets held firm for a famous victory which boosts their chances of avoiding relegation.
Barclays Prem
1 Chelsea 63
2 Man City 58
3 Arsenal 57
4 Man Utd 53
5 Liverpool 51
6 Tottenham 50
7 So'ton 49
8 Stoke 42
9 Swansea 40
10 West Ham 39
11 Newcastle 35
12 C Palace 33
13 W Brom 33
14 Everton 28
15 Hull City 28
16 A Villa 28
17 S'land 26
18 Burnley 25
19 QPR 22
20 Leicester 19
Earlier, Olivier Giroud's thunderbolt helped Arsenal see off the challenge of West Ham and cement their place in the top four of the Barclays Premier League with a 3-0 victory.
The Gunners recorded an eighth home league success to keep the pressure on second-placed Manchester City after finally seeing off what was a spirited second-half effort from the Hammers.
Theo Walcott, recalled to the starting XI following injury to Alex Oxlade-Chamberlain, had missed a couple of early chances before goalkeeper Adrian was beaten on the stroke of half-time by a fierce angled drive from French striker Giroud, which cannoned in off the far post.
West Ham, who have plummeted down the table following a poor run of results, regrouped for the second half and would have been good value for an equaliser.
However, Arsenal finally broke then down again with nine minutes left when Aaron Ramsey fired in after a neat one-two with Giroud and substitute Mathieu Flamini rolled home a third to send Arsene Wenger's side into next week's Champions League tie away to Monaco with renewed hope of an unlikely comeback.
Christian Benteke and Gabriel Agbonlahor combined to leave Gus Poyet's job hanging by a thread as Aston Villa thrashed Sunderland 4-0 at the Stadium of Light.
Villa, who had scored only four Barclays Premier League goals on the road all season, doubled their tally inside 44 remarkable minutes, and but for a glaring miss by Scott Sinclair, could have led 5-0 at the break.
However, doubles from Benteke - his first ended a 614-minute wait for an away goal in the league - and Agbonlahor gave them a commanding lead they never looked like surrendering as they ran out easy winners.
That left Poyet, who was confronted by angry supporters close to his dug-out, to walk the gauntlet as the locals among a crowd of 45,746 - the biggest of the season - at the Stadium of Light, voted with their feet with many leaving long before the final whistle.
Villa's third successive win, two of them in the league after seven defeats on the trot, eased Tim Sherwood's men further clear of the drop zone, but left Sunderland deep in trouble and the Uruguayan, whose side has now won just one of its last 11 league games on home soil, facing a no-holds-barred inquest with the Black Cats' season unravelling alarmingly.
Brown Ideye's first-half winner sealed a narrow 1-0 victory for West Brom as Tony Pulis returned to haunt Stoke again.
The striker's seventh goal of the season settled the game to move the Baggies 11 points clear of the Barclays Premier League relegation zone.
Barclays Prem
W Brom 1 - 0 Stoke
Burnley 1 - 0 Man City
S'land 0 - 4 A Villa
Leicester 0 - 0 Hull City
C Palace 3 - 1 QPR
Arsenal 3 - 0 West Ham
Ex-Potters manager Pulis - who also beat Stoke 1-0 while at Crystal Palace last season - saw Albion recover from back-to-back defeats against Aston Villa.
They were also denied a penalty after Craig Dawson appeared to be fouled in the box and Ideye had a goal disallowed.
Bottom club Leicester failed to see off Hull after Tom Huddlestone's sending-off handed them a huge opportunity at the King Power Stadium.
With both sides struggling at the wrong end of the table, this laborious 0-0 draw lived down to expectations, but the Foxes had both home advantage and the greater need for points and will be frustrated not to have gathered a desperately-needed win.
They were second best for most of the game, but when Huddlestone was dismissed for a second booking in the 71st minute they had a priceless chance to redefine their faltering campaign.
Manager Nigel Pearson, a less-than-beloved former Hull boss, emptied his bench of attackers in pursuit of three points but the stalemate endured.
Tigers skipper Michael Dawson was outstanding at the back, marshalling Hull's defence throughout, and without his organisation they might well have creaked under pressure.
In the lunchtime fixture, three first-half goals helped Crystal Palace increase QPR's relegation fears after a 3-1 victory at Selhurst Park, where the only positive for the Rs was a contender for goal of the season from Matt Phillips.
Wilfried Zaha opened the scoring after meeting a low Yannick Bolasie cross at the back post, and painfully collided with the post for his troubles.
The Rs were in disarray at the back, and Bolasie again was allowed too much room to cross for James McArthur to double his side’s lead five minutes before half-time.
Then just three minutes later Joel Ward notched a rare goal to complete a miserable opening 45 minutes for QPR.
Julian Speroni had one of his least eventful afternoons of the season in the Palace goal, until Phillips notched an incredible strike from 45-yards out with ten minutes left.
The result sees Chris Ramsey’s side face an even tougher challenge to pull themselves out of the relegation mire, while Palace pulled further clear of the bottom three, and now sit comfortable in 12th.