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.
But Akinbiyi headed in Mike Duff's cross to level and secure a point.
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.
Att: 10,659.
Ref: E Ilderton (Tyne & Wear).
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