The Gunners had won their last 11 meetings with Burnley in all competitions but Mikel Arteta's men did not do nearly enough to extend that run as they again failed to make up significant ground on the top four.
Although Arsenal spurned presentable chances of their own, they could consider themselves a little fortunate to pick up a point as Jay Rodriguez somehow contrived to hit the underside of the bar from five yards out with 12 minutes remaining, the ball coming down on the goalline.
Both teams remain on 31 points, 10 adrift of the top four and seven above the bottom three.
An unmarked Alexandre Lacazette headed wastefully off-target from Pierre-Emerick Aubameyang's cross as Arsenal started brightly, before the latter latched on to David Luiz's lofted ball over the top, only to screw a shot hopelessly wide of the near post.
Burnley were grateful for a superb last-ditch tackle from James Tarkowski on Lacazette when the forward looked set to score, but the hosts soon grew into the game as Arsenal's threat faded.
Either side of Nick Pope staying big to deny Aubameyang, Rodriguez's low 25-yarder tested Bernd Leno while the impressive Dwight McNeil and Jeff Hendrick each went close.
Burnley certainly carried the greater threat early in the second half, Tarkowski and Ben Mee both threatening from set-pieces prior to Jeff Hendrick heading a McNeil cross narrowly past Leno's right-hand post.
Aubameyang sent a header inches past the post from Lacazette's left-wing delivery in a rare moment of second-half danger for the hosts, but it was Burnley who came closest to a winner, Rodriguez failing to convert McNeil's knock-down when it looked easier to score.