Wilfried Zaha and Jeffrey Schlupp grabbed the goals as Roy Hodgson’s team picked up three welcome points.
The Clarets, who brought in Robbie Brady for the suspended Ashley Westwood, made a solid start to the match although the only half-chance in the opening exchanges fell to Jack Cork, whose 25-yard drive was easily gathered by Vicente Guaita in the Eagles’ goal.
Palace, who started with Martin Kelly and Scott Dann in defence as Joel Ward and Gary Cahill were injured, were on the back foot in the first half and Guaita was forced to keep out Dwight McNeil’s drive.
Burnley suffered a blow when Charlie Taylor had to be substituted after picking up an injury, while Jordan Ayew rightly had a goal ruled out for offside.
A very dreary first half only sporadically burst into life but Palace took the lead against the run of play in stoppage time when Zaha’s left-footed shot proved to be too good for Burnley goalkeeper Nick Pope.
Sean Dyche’s team started the second half well and Erik Pieters fired the ball into the side-netting. At the other end, Ayew sent his shot wide after an error by James Tarkowski.
Chris Wood lashed a presentable opening high and wide from just six yards out before James McArthur cleared another effort by the New Zealand international off the line.
Burnley poured forward in search of an equaliser but Palace held firm and added a second goal with 10 minutes remaining when Schlupp beat Pope after an error by Ben Mee.
McNeil and Wood went close to scoring for the home side in the closing stages but Palace showed once again why they tend to pick up more points on their travels than at Selhurst Park.