Chris Eagles hit the crossbar and Palace goalkeeper Julian Speroni produced a hat-trick of great saves to deny young striker Jay Rodriguez, who also saw a shot come back off the angle during the closing stages on Saturday.
Palace made a bright start and Burnley were almost caught out as early as the fourth minute. Alex Marrow managed to get in behind Danny Fox and fired a pacey ball across for Kieran Djilali, but the midfielder blazed over Brian Jensen's crossbar.
Djilali then turned provider, crossing for Neil Danns who should have done better with a header which dropped into the arms of Jensen.
Ross Wallace hobbled off with 18 minutes gone and was replaced by Rodriguez.
Temporarily down to 10 men, Eagles did well to get back and make a vital interception as James Vaughan looked to hook the ball back into the danger zone.
Jensen then comfortably tipped over a speculative effort from Andy Dorman in another worrying moment for the Clarets.
The impressive Eagles almost gave the visitors the lead in the 25th minute when he got away from two defenders and clattered the crossbar from 30 yards with home goalkeeper Speroni well beaten.
Dean Marney skied over the bar shortly after and Speroni made a fine save to his left-hand side after Rodriguez read Tyrone Mears' clever back heel and drove a shot towards goal from the edge of the box.
A neat interchange of passes between Eagles, Chris Iwelumo and Wade Elliott resulted in Rodriguez emerging with the ball inside the box, but Speroni made a superb save as the youngster looked to find the net.
Eagles, who was running the show, cut inside, slid past Marrow and fizzed a shot just over the bar, while Rodriguez had a volley saved by Speroni and Marney skewed just wide as the away side began to dominate proceedings.
There was a moment of danger at the other end. Wilfried Zaha sprang the offside trap and burst clear but Jensen came to the Clarets' rescue with an excellent block.
Steven Thompson came on for Iwelumo in the 70th minute but it was the hosts who were on the offensive.
Palace always looked dangerous on the break and had Mears not been so alert with 15 minutes remaining Danns' goalbound shot might have caused Jensen a few problems.
Rodriguez then hit the angle after a Mears pull-back, capping a miserable afternoon for the 21-year-old who could have finished with a hat-trick.