The Blues were temporarily pegged back when Ashley Barnes diverted the ball past Thibaut Courtois, cancelling out an own goal from Kevin Long, only for Victor Moses to hit back and earn his side a deserved victory.
On a night of rotation for Antonio Conte, striker Alvaro Morata's terrible miss at 1-0 looked as though it could cost his side at one point, but the visitors showed good character to leave Lancashire with the win needed to keep their top-four hopes alive.
Chelsea, who had a weak penalty appeal rejected early on when Olivier Giroud went to ground inside the box, were on top from the off and should have been ahead prior to the own goal.
Some nice interplay saw Morata race in behind, only for the angle to work against him as Nick Pope stuck out a leg to keep out the shot.
The opener arrived with 20 minutes on the clock and was created by Moses, whose cross from the right was bundled over the line by Long after the smallest of touches from Pope on the initial delivery.
Chelsea, with just one win in 21 to their name this calendar year heading into this fixture, continued to see more of the ball and looked the more dangerous side as half time approached.
Morata was thwarted again by Pope 37 minutes in after being played in by Giroud, falling short with his delicate finish, and N'Golo Kante drilled over the bar when working some space right on the edge of the box.
Despite being without the likes of Eden Hazard, Willian and Cesc Fabregas, each of whom were left on the bench in a side showing six changes, the visitors should have been two in front come the hour mark.
Morata had half the length of the field to run into on the counter, but the Spaniard could only place the ball wide of Pope's target when one on one, explaining his angry reaction to being hauled off later in the match.
A slightly trickier chance for Moses ended with Pope producing a routine stop, and Chelsea's inability to put the ball away would prove costly.
Against the run of play, and with a huge slice of luck, Burnley levelled things up as Johann Berg Gudmundsson's shot hit the heel of Barnes and crept past Courtois.
Parity was restored for just five minutes, however, after Moses fired the ball past Pope at his near post moments after the English stopper had kept out Emerson Palmieri's drive.
Sean Dyche turned to Sam Vokes and saw the Welshman get a headed attempt on target 30 seconds after his introduction, though the Clarets - who would have climbed above Arsenal into fifth with three points - could not create any further killer chances.
BURNLEY (4-4-2): Pope; Lowton, Tarkowski, Long, Ward; Lennon (Wells 87'), Westwood, Cork, Gudmundsson; Wood (Vokes 72'), Barnes
CHELSEA (3-5-2): Courtois; Azpilicueta, Cahill, Rudiger; Moses, Kante, Bakayoko, Pedro, Emerson (Zappacosta 84'); Morata (Hazard 71'), Giroud