With seconds of the game remaining, a corner bounced off defender Laurent Koscielny -- with Burnley appealing that the ball had hit his hand -- and went in at the far post after Theo Walcott and Alex Oxlade-Chamberlain had got touches.
Both sides had come close in the final quarter of an hour, with the Gunners' Alexis Sanchez striking the outside of a post and Michael Keane smacking a header off the Arsenal bar.
Arsenal had the first chance to break through in what turned out to be a low-key first half, with an angled low shot from Sanchez taking a touch off teammate Walcott and rolling wide of the far post in the fourth minute.
Mesut Ozil then scuffed an effort harmlessly, but Burnley were growing into the game and had a chance of their own after 16 minutes when Jeff Hendrick found George Boyd on the left but, with players in support, Boyd elected to shoot and drilled the ball wide of Petr Cech's goal.
Neither side could find the right final pass, with Walcott skewing one attempt across the area before Santi Cazorla, in a promising crossing position, failed to steer the ball past the first Burnley defender.
Ozil slipped just as it seemed he would have the chance to shoot from just outside the area before Burnley made -- and missed -- the best chance of the match so far with five minutes to go until the break.
Matt Lowton's cross from the right found Sam Vokes unmarked and onside but, from a great position, the big striker powered a header wide of the target from 10 yards out.
Lowton was again involved at the start of the second period when he steered another cross towards Vokes, who was unable to make clean contact -- but Arsenal stepped up the pace and Sanchez had an effort smothered by keeper Tom Heaton before Walcott lashed over.
With 53 minutes on the clock, Alex Iwobi was found in space on the left and cut inside the area before ballooning a woeful effort miles off target as the visitors again failed to make the most of a promising situation.
And shortly before the hour mark Burnley came agonisingly close making the breakthrough when Petr Cech made a fine one-handed stop as Johann Berg Gudmundsson's firm downward header from a Stephen Ward cross threatened to find its way into the corner.
From the resulting corner, Arsenal had to hack desperately clear after the ball bounced around in a bout of goalmouth pinball.
As the game ticked into its final quarter of an hour, both teams came within a whisker of the opener.
First, a thumping Sanchez strike brushed the outside of the post on its way wide, and then Burnley's Keane clattered a header against the Arsenal bar.
Five minutes from the end, Walcott curled narrowly wide of the far post with Heaton scrambling after Sanchez had won the ball inside the Burnley area.
Oxlade-Chamberlain saw a last-ditch effort deflected over the top -- but within seconds, Koscielny snatched the points from the corner. PLACE YOUR TEXT ADVERT BELOW:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>