With just 24% possession and a single shot on target, which arrived inside the first 10 minutes, the Clarets were made to defend doggedly in North London.
But they went about their business rather diligently against the Gunners and restricted the hosts to just five shots on target across the 90 minutes.
Nick Pope saved with his legs to deny Martin Odegaard and preserved a fourth clean sheet of the campaign with a fine stop to keep out Emile Rowe Smith's effort inside the box.
Alexandre Lacazette somehow missed the target late on when nipping ahead of Ben Mee to meet Smith Rowe's pull back, but the French striker flicked the ball wide of the upright.
Mikel Arteta's side, however, didn't have it all their own way as Dwight McNeil threatened on several occasions.
The ex-England Under 21 international's cross-shot was tipped over the bar by Aaron Ramsdale in the first half, the winger's low effort was turned for a corner in the second while a late attempt just cleared the crossbar.
Here are the ratings.
5. Ben Mee 8
Allowed Lacazette on the outside when the Frenchman skewed an effort wide in the first half, but made a good block to keep out the striker's rebound moments later. Didn't lose a header in his own penalty area and hooked an early effort just wide of the upright. Photo: Glyn Kirk
6. Aaron Lennon 6.5
Pressed well from the front, hurrying Tierney on the ball in his own third, and tracked the Scotsman's runs when getting forward in support of Martinelli. Always alert to the threat of the Gunners pair, though he struggled to make an impact in possession. Photo: Julian Finney
7. Dwight McNeil 8
Offered much more on the ball against the Gunners, threatening on the break while easing the pressure on the edge of his own penalty area with some neat touches. Two shots on target, both tested Ramsdale at his near post, and he cleared the crossbar late on. Photo: George Wood
8. Josh Brownhill 7
Grafted in the middle of the park, competing well for the ball and breaking up play. Forced to cover plenty of ground as Arsenal broke through the lines, with Smith Rowe and Odegaard getting goalside, but recovered to provide a layer of protection. Booking means he'll miss Watford. Photo: Glyn Kirk