Burnley gifted the visitors the lead with some poor defending that has been a feature of previous home games.
A long clearance by Palace keeper Gabor Kiraly caused panic in the centre of Burnley's defence, allowing Clinton Morrison to head over hesitant goalkeeper Danny Coyne.
Burnley almost equalised immediately as Andy Gray was denied by a goal-saving tackle from centre-half Leon Cort.
The Clarets finally sprang into life when Akinbiyi tested Kiraly, who made a rare save by tipping the ball over for a corner.
It was a surprise when Burnley equalised before half-time as full-back Michael Duff crossed for Akinbiyi to tie things up with a classic header.
Burnley continued to make the running early in the second period with Gray collecting a pass from Chris McCann and bringing a save from Kiraly with a shot on the turn.
Akinbiyi should have had his second goal of the game shortly afterwards but headed tamely at Kiraly.
The former Leicester City striker then had another opportunity after being sent clear by Duff but fell between shooting at the target and trying to find fellow striker Gray.
A further chance went begging for the home side as Jon Harley headed a cross from Wade Elliott wide.
Palace boss Peter Taylor made a double substitution by introducing Lewis Grabban and Ben Watson in an attempt to stem the Clarets tide.
The changes appeared to work as Palace created a couple of late opportunities with Morrison turning a cross from Danny Butterfield just wide of the far post and midfielder Tom Soares firing just over from 20 yards.