A goal in each half from James McArthur and Andros Townsend condemned Sean Dyche's side to their seventh game without a victory but the scoreline barely reflected the dominance of the home side.
Palace were superior in every department, moving the ball at tempo and attacking with intent while Burnley looked like a team utterly lacking belief.
McArthur put Palace, who had been without a home win, ahead in the 16th minute with a low cross-shot which beat a stationary Joe Hart and went in off the post.
Wilfried Zaha caused the Clarets defence plenty of problems, forcing two saves out of Hart early in the second half before he crashed a shot against the crossbar.
Palace skipper Luka Milivojevic wasted a great chance when the ball fell to him alone in the box but he screwed his shot wide.
Burnley, 18th in the table and looking short of confidence and ideas, created little and Palace finally got their second in the 77th minute.
Townsend cut in from the right flank on to his left foot and beat Hart with a magnificent shot from outside the box which flew into the top corner.
"I’m delighted because we played very well, we have had teams on the rack at home and not got the win we needed, so it was good to get a win after a good performance," Palace manager Roy Hodgson said.
"I didn’t think we were going to relinquish our grip and let them back in. We did what we needed to do and I am delighted we won it with such a good goal," added the former England manager.
While Palace move up to 14th place with 12 points, Burnley look likely to face a long battle against the drop after enjoying a seventh-placed finish last season.
"The best side won, simple as that. We are in the position where the eye of the tiger has gone, last season everyone was on it," Dyche said.
"To get the eye of the tiger back, a boxer has to get back in the gym. We’ve got to get back on the training ground and work hard.
"It's frustrating, they have given so much to the cause but then fear comes in, it’s a tough thing".
(Reporting by Simon Evans, editing by Ed Osmond)
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