Palace looked likely to pay the price for letting a host of chances go begging when Mike Dean pointed to the spot in the 86th minute, but Scott Arfield’s well-struck penalty was brilliantly parried to safety by the Eagles keeper.
Scott Dann hit the bar for Neil Warnock’s strugglers in the first half, while Wilfried Zaha, Jason Puncheon and Damien Delaney also went close before Burnley brightened after the break and looked the stronger side in the closing stages.
While both sides remain without a Premier League win this season, their work-rate, spirit and attacking intent could not be questioned from the outset.
Danny Ings felt he had claims for a penalty in the fourth minute after skipping past Delaney and going to ground on the edge of the box, only to be brandished a yellow card for diving by Dean.
Delaney then spurned Palace’s first clear-cut chance, collecting Zaha’s incisive cut-back before turning sharply and firing inches over from 15 yards.
A minute later, the home side were denied by the woodwork as Frazier Campbell’s floated cross to the far post picked out Dann, whose header beat Tom Heaton but not the crossbar.
Palace then had their own penalty claims waved away by Dean when Jason Shackell misjudged the flight of Puncheon’s cross from the right flank and appeared to bring the ball down with an arm before belting it upfield.
Zaha was a constant threat to the visiting defence, and his storming run and shot was parried by Heaton onto Shackell’s shin before the Burnley skipper reacted quickly to hoof the goal-bound ball away to safety.
The visitors looked dangerous on the counter-attack and Arfield scooped a half-volley over the bar from Kieran Trippier’s near-post centre, but the visitors then lost the services of Ings after he pulled up clutching his hamstring and limped off to be replaced by Marvin Sordell.
Palace went back on the offensive and Puncheon scuffed a disappointing strike straight at Heaton after neat link-up play with Dwight Gayle, who then blazed high and wide after finding himself in acres of space on the edge of the box and rushing his shot.
Another flowing Burnley counter-attack presented a chance for George Boyd, but Dann flung himself in front of the shot and diverted the ball behind in the last meaningful action of the opening 45 minutes.
Palace continued to look the more likely to break the deadlock after the interval as Gayle came within inches of getting his rainbow laces to Mile Jedinak’s header across goal.
Delaney then glanced a near-post header into the side-netting, and Puncheon did likewise from a tight angle after being teed up by a neat chest-down from Campbell.
Record signing James McArthur came within 18 inches from marking his Palace debut with a goal, but his rasping 25-yarder was still rising as it disappeared into the Holmesdale Stand.
But Sean Dyche roused his troops and they responded by creating their best chance of the game, Arfield given room to roam forward before letting fly with a thundering 20-yard strike that Speroni did well to palm over.
The home keeper was forced into action again moments later as he got down sharply to his right to intercept a low drive from David Jones, and the lively Arfield then cut inside Joel Ward only to curl his shot high and wide.
Jonny Williams almost made an instant impact after replacing Gayle, but the diminutive Welshman stabbed a hurried effort wide of the post following excellent work from Yannick Bolasie.
But with five minutes remaining, Jedinak was adjudged to have wrestled Lukas Jutkiewicz to the floor and referee Dean had no hesitation in awarding the penalty.
Up stepped Arfield, whose powerful spot-kick looked destined for the top corner until Speroni launched himself to his right and beat the ball away with a firm left palm, and the alert Palace defender Adrian Mariappa won the race to the loose ball.
The home defence comfortably dealt with a pair of late corners, and both sides settled for their second points of the season at the end of a combative contest.