Gareth Taylor headed Burnley in front three minutes before the break before Marlon Beresford spoiled an otherwise immaculate display by giving away a penalty after a rash challenge on Dele Adebola in the 61st minute.
Beresford, who is on a month to month contract at Turf Moor, made a number of vital saves to keep his side's goal intact, before Taylor did the business at the other end with a well-placed header from Robbie Blakeâ€™s inswinging free kick from the left byline.
Earlier Julian Gray was the unlucky victim of some outstanding saves from Beresford. Aki Riihilahti missed an easy opportunity and Tommy Black fired into the side-netting from just five yards out as the Eagles enjoyed the early play.
But thanks to Beresford and some luck the Clarets held out and then delivered a sucker-punch just before the break as Taylor netted his tenth goal of the season.
Palace came out with all guns blazing after the break but it took a long ball from Hayden Mullin, chased by Adebola, to put the Eagles back into it.
Beresford came racing out of his goal to the edge of the area to send the big Palace striker sprawling.
The former Middlesbrough keeper was perhaps lucky to just pick up a yellow card as Popovic kept his cool and hoisted the spot kick into the top right-hand corner of the net.
Tommy Black then hit the bar and Beresford denied Ade Akinbiyi while at the other end Robbie Blake hit the post.
Palace did all the pressing as they strived for all three points to lift them into the top six but Burnley held out with stout defending and could even have nicked all three points had Driss Diallo kept his ten-yard shot down and not lifted it over the bar.