Crystal Palace secured a deserved point at Turf Moor with a gritty display.
Paul Dickov volleyed narrowly wide from 18 yards before Mark Hudson headed in Ben Watson's whipped-in set-piece to give Palace the lead on five minutes.
The Clarets hit back when Michael Duff shot home from six yards after good work on the left by Robbie Blake.
Burnley's Alan Mahon drilled an effort wide from 25 yards after the break while Watson went close with a long-range strike.
Burnley manager Steve Cotterill:
"In the second half we were a different team and I was confident we would win the game.
"This is a tough old league and if someone gets their nose in front it's really difficult.
"The players deserve a lot of credit - everyone will think I tore a strip off them but they showed great quality to come back and should have won."
Crystal Palace manager Peter Taylor:
"I think to be fair it was an interesting game but both teams conceded at set pieces.
"The players worked hard for 90 minutes and this is not a bad point as Robbie Blake is in such good form, so I think we did well."
Burnley: Kiraly, Duff (Spicer 39), Carlisle, Caldwell, Jordan, Elliott, Alexander, Blake, Mahon, Gray (Jones 70), Lafferty (Akinbiyi 70).
Subs Not Used: Jensen, Unsworth.
Goals: Duff 13.
Crystal Palace: Speroni, Butterfield, Hudson, Cort, Craig, Soares, Fletcher, Watson, Green (Hughes 60), Scowcroft (Morrison 76), Dickov (Idrizaj 60).
Subs Not Used: Flinders, Lawrence.
Goals: Hudson 5.
Ref: Trevor Kettle (Rutland).