The result leaves both teams on 31 points after 25 games of the campaign with the 10th-placed Gunners only ahead of Sean Dyche’s side on goal difference.
Arsenal, who welcomed Pierre-Emerick Aubameyang back from suspension and had the returning Matteo Guendouzi in their line-up, made the better start and, after only two minutes, Alexandre Lacazette should have done better with a header.
Aubameyang had a good opportunity to break the deadlock in the 14th minute when he latched on to a long ball forward from David Luiz but then conspired to miss the target from 10 yards out.
Burnley, unchanged from the side that beat Manchester United at Old Trafford in their last Premier League match, created their first opening five minutes later when Chris Wood laid the ball off to Jay Rodriguez, whose low 25-yard effort was tipped around the post by Bernd Leno.
In an even first half, Guendouzi’s long-range shot was deflected wide and Shkodran Mustafi headed over the bar for Mikel Arteta’s side while Dwight McNeil went close with a fierce 30-yard effort and an unmarked Jeff Hendrick fired wide for the Clarets.
After the break Mesut Ozil found Gabriel Martinelli who had a shot blocked while, at the other end, James Tarkowski and Rodriguez both headed presentable chances wide.
As Dyche’s side enjoyed a purple patch midway through the second half, Hendrick failed to find the target with a header from McNeil’s excellent cross and Rodriguez dragged another shot wide.
When Arsenal got forward Aubameyang contrived to send his 10-yard header wide while Rodriguez should have won the game for Burnley only for his close-range volley to come off the underside off the bar but fail to cross the line.
Both teams now have a weekend off before returning to action in two weeks’ time.