Wilfried Zaha put the Eagles ahead in first-half stoppage time, with substitute Jeffrey Schlupp sealing an excellent win in the closing stages.
Story of the match
Palace increasingly grew into the game as the first-half progressed and briefly thought they had grabbed the lead on 38 minutes.
James McArthur slid Patrick van Aanholt through down the left, and he cut the ball back to Andros Townsend, who looped in a cross. It eventually broke for Jordan Ayew, who fired home, only for Van Aanholt to be correctly ruled offside when he intervened in the scramble.
They would, however, find a breakthrough before the interval courtesy of Zaha.
The forward picked up the ball on the left-hand side and ran at Phil Bardsley before fashioning shooting room from a tight angle and squeezing it past Nick Pope at his near post with his weaker foot.
13 minutes into the second period, Palace very nearly handed the hosts an equaliser.
Scott Dann headed clear to Cheikhou Kouyate from a free-kick, but the Senegalese midfielder inexplicably headed it back towards his own goal and Chris Wood latched onto it. One-on-one with Vicente Guaita, the usually prolific frontman ballooned his effort over the top, leaving the Turf Moor faithful stunned.
Shortly after, on the hour mark, they were ready to celebrate again after Ben Mee connected with Jeff Hendrick's corner and Wood seemed to destined to finish, but McArthur managed to hack the ball off the line.
Then, on 78 minutes, Clarets skipper Mee made a decisive error, fumbling a clearance. Substitute Schlupp pounced, his powerful low drive finding the net via Pope's leg.
Burnley looked to spark a frantic finish but Jack Cork spurned their best chance, awkwardly lifting the ball over the top after Guaita had made an excellent save from Wood's header.
Palace back to winning ways as schedule begins to look kinder
This is the first time Roy Hodgson's men have won in the Premier League since their 2-1 victory at West Ham United at the start of October.
Since then, they have been beaten by every side in the current top four and drawn at Arsenal. It has been a gruelling run.
They can be expected to open up a larger cushion between themselves and the relegation zone, especially with Zaha having found his shooting boots once again.
Missed opportunities for Clarets
Burnley have a tendency to hoover-up points when sides outside the 'big six' visit Turf Moor, and this has proven essential in keeping them out of trouble.
This result can probably be put down to a blip given the usual consistency of Sean Dyche's side, but still there will be great frustration.
They'll look to produce a response when reigning champions Manchester City come to town in midweek.