The hosts looked like slipping towards the drop zone after Clinton Morrison was gifted an opening goal, looping a header under the bar early on.
The striker had chances to win it but twice missed the target, while Lewwis Spence went close for Palace late on.
Crystal Palace manager Peter Taylor:
"We finished the game well. I thought we had a couple of quite good chances in the end and I'm very pleased with the way we finished.
"Burnley away is a difficult game and I'm pleased with an away point.
"I'd have to look at the video but it was probably a very disappointing goal for us to concede."
Burnley: Coyne, Duff, McGreal, Caldwell, Harley, Elliott, James O'Connor, Gudjonsson, McCann (Jones 80), Gray (Garreth O'Connor 89), Akinbiyi.
Subs Not Used: Djemba-Djemba, Spicer, Foster.
Booked: Gudjonsson, James O'Connor.
Goals: Akinbiyi 38.
Crystal Palace: Kiraly, Butterfield, Hudson, Cort, Lawrence, Soares, Fletcher, Spence (Watson 66), Freedman (Grabban 63), Morrison, Scowcroft.
Subs Not Used: Speroni, Ward, Martin.
Booked: Scowcroft, Watson.
Goals: Morrison 15.
Ref: E Ilderton (Tyne & Wear).