A goal from Junior Stanislas nine minutes from time sent Palace to only their third defeat in 25 games after the home side had spent most of the second half on the back foot.
Ian Holloway named new loan signing Alex Nimely on the bench as his side returned to league action for the game at Turf Moor, while the home side were without strikers Charlie Austin and Martin Paterson, both out with hamstring injuries.
An early chance came Palace's way as they broke clear after a Burnley corner and the ball fell to Murray in the box but his finish went just inches wide of Lee Grant's post.
Burnley replied in the right way with two chances of their own as Sam Vokes was put through by Kieran Trippier but shot over the bar. Stanislas then found Danny Ings but he wasn't able to keep the ball down and also didn't trouble Julian Speroni.
Wilfried Zaha showed his threat on the left hand side of the box as he cut inside to beat his man and with his shot looking destined for the net but was pushed around the post by keeper Lee Grant just past the 30 minute mark.
At the other end Ings headed wide from a Stanislas cross and Speroni saved on the line following a touch from close range by the same player.
If it were not for their goalkeeper, Burnley would then have fallen behind as Grant did brilliantly to turn the ball away after Jason Shackell had diverted Jonathan Parr's cross towards his own goal.
A good start to the second period by the visitors saw a couple of early chances. Dean Moxey curled a shot just wide with his right foot that had Grant scrambling to his right and then Yannick Bolasie crossed from the right and found Murray but the striker couldn't get enough power behind his header towards goal.
Just before the hour mark of the game a fine move from the home side saw Ross Wallace find Trippier who put in a fine cross that was met by Marney at the far post but Speroni saved the header.
The visitors were having the better of the play and the Clarets were thankful to a good block by Edgar as he cut out low ball into the six yard box from Moxey. A Mile Jedinak effort hit the wall but Zaha created space for himself from the rebound but his shot was pushed away by Grant.
In probably Burnley's first chance of the second half Ings got the better of his man on the right and found Stanislas on the edge of the box who calmly controlled the ball and curled it into the top corner of the net past Speroni to break the deadlock.
Five minutes of added time saw two further Speroni saves from Vokes as the home side nearly increased the advantage and Mile Jedinak's shot hit the woodwork but the day ended with no return from their travels for Holloway's side.
Burnley: Grant, Trippier, Lafferty, Duff, Shackell, Edgar, Marney, Ings, Wallace (Bartley 82), Vokes, Stanislas (Treacy 82).
Subs not used: Richards, O’Neill, MacDonald, Long, Jensen.
Booked: Treacy 90
Crystal Palace: Speroni, Parr, Ramage, Delaney, Moxey, Jedinak, O’Keefe (Nimely 82), Bolasie (Wilbraham 90), Zaha, Murray, Moritz (Williams 69).
Subs not used: Easter, Parsons, Gabbidon, Price.
Referee: Gavin Ward. Att: 11,564