A goal from Bakary Sako was enough to extend Burnley’s winless run after a disappointing performance from Sean Dyche’s side.
The home side were by far the better side in the first-half as they continue to improve under former-England manager Roy Hodgson, putting the Clarets under early pressure
Burnley were lucky not to concede a penalty when Bakary Sako went down under a challenge from James Tarkowski who appeared to miss the ball and make contact with the pacey winger.
But Palace continued to ask questions of the Burnley defence and soon took the lead through Sako who found his way into the penalty area far too easily and powered past Nick Pope in the Burnley goal.
And the hosts almost doubled their lead when a weak pass from Charlie Taylor allowed Wilfried Zaha to flick the ball over Pope towards James McArthur, only for Ben Mee to intervene and clear the danger superbly.
Palace again went close to scoring a second on the verge of half-time as Christian Benteke lost his man with ease and powered over the bar from close-range following a corner from Luka Milivojevic.
The Clarets created eventually got a chance on goal when Sam Vokes’ glancing header dropped narrowly wide of Wayne Hennessey’s far post after a cross from Johan Berg Gudmunsson.
And they away side again threatened to equalise against the run of play as Ashley Barnes forced Hennessey into action as the Welsh goalkeeper tipped a dipping effort over the bar.
The home side came close on the verge of half-time when Sako was inches away from a second when his thunderous volley flashed wide of the far post after a cross form Timothy Fosu-Mensah.
Burnley improved in the second-half and nearly equalised when Vokes again met a Gudmunsson, but the Welshman could only find the side-netting as he flicked on at the near post.
Loanee Georges-Kevin Nkoudou made his Clarets debut and made an immediate impact, threatening the Palace defence with his pace and dribbling ability.
Palace threatened through Benteke who nodded into the side netting after rising well at the near post, but they failed to seal the win.
The Clarets had one last chance to salvage a point when Barnes was denied by a smart save form Hennessey after a threatening pass from Steven Defour.
Player Ratings
Crystal Palace: Hennessey (7) Fosu-Mensah (7) Kelly (7) Tomkins (6) Van Aanholt (6) Zaha (7) Milivojevic (7) Riedewald (6) McArthur (7) Benteke (6) Sako (8)
Substitutes: N/A
Burnley: Pope (6) Bardsley (5) Tarkowski (6) Mee (7) Taylor (6) Cork (6) Defour (6) Gudmunsson (6) Hendrick (5) Barnes (6) Vokes (6)
Substitutes: Nkoudou (6) Wells (N/A)
Man of the Match: Bakary Sako
It is now seven league games without a win for Burnley after a disappointing performance, although they remain in seventh place with Leicester three points behind.
They host Manchester United on Saturday in their next Premier League fixture, looking to repeat their superb performance at Old Trafford on Boxing Day which saw them earn a 2-2 draw.
ClaretDan 