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.
9. Ashley Westwood 7
Continued to look for ways to unlock the Arsenal defence by dropping passes alongside Gabriel and Holding. An energetic performance, though couldn't prevent Lokonga from dictating from deep and could perhaps consider himself fortunate when escaping with a booking for a foul on Tierney. Photo: Catherine Ivill
10. Jay Rodriguez 6
A combative display from the striker in the final third, winning flick-ons, challenging for second balls and pressing the centre backs. Quietened down after the break, but kept Westwood's ball alive when McNeil forced Ramsdale into a save at his near post. Photo: Catherine Ivill
11. Matej Vydra 6
Put himself around, didn't give Arsenal's rearguard any time to play out from the back, and did the dirty work when Arsenal cranked up the pressure late on. A bright spark in the air, but lacked composure in key areas and decision making wasn't up to scratch at times. Photo: Catherine Ivill