Paul Dickov had the game's first opportunity when he latched on to a header from James Scowcroft only to shoot inches wide.
But Palace didn't have long to wait before taking the lead when a Ben Watson free-kick was headed home by centre-half Mark Hudson.
Dickov had another chance when he intelligently let the ball run through to Scowcroft, took the return pass but shot straight at former Palace keeper Gabor Kiraly.
Burnley equalised on 13 minutes as a Robbie Blake free-kick appeared to evade a whole cluster of players and Argentinian goalkeeper Julian Speroni could only palm the ball into the net.
Burnley were forced to make a change when right-back Duff was forced off with an injury.
After 44 minutes Kyle Lafferty tried his luck from 25 yards and was unfortunate to see his shot curl just wide.
The half finished with Kiraly and Dickov colliding on the edge of the box with the ball falling to Scowcroft, whose shot was cleared off the line by Clarke Carlisle.
Speroni was twice called into action in the early stages of the second half saving well from Andy Gray and Alan Mahon.
Watson responded for Palace by jinking around two tackles and firing just wide.
On 58 minutes Burnley caught the visitors' defence asleep as substitute John Spicer's pass was turned over the crossbar by Gray from inside the six-yard box.
Seventeen minutes from time Blake missed a glorious opportunity to find the net as he created a shooting chance for himself but shot just too high.
Burnley came even closer in the last minute as substitute Ade Akinbiyi chested the ball down for Wade Elliott, but the winger's shot cannoned back off the crossbar.