Martin Odegaard's sumptuous set-piece on the half-hour separated the sides and inflicted a fourth defeat from five on the Clarets.
The Norwegian international swept a free-kick beyond Nick Pope's reach after Ashley Westwood had upended Bukayo Saka on the edge of the box.
Burnley were much better after the break with the introductions of debutant Maxwel Cornet and Matej Vydra giving the hosts a shot in the arm.
The latter thought he'd won his side a penalty when going to ground under Aaron Ramsdale's challenge.
However, following the Gunners' protestations, referee Anthony Taylor overturned the decision after checking his monitor.
Cornet also made his mark, giving the home support a significant lift, and he also marked his first appearance for the club with a goal.
The ball broke for the Ivorian inside the penalty area, but Ramsdale turned the attempt over the crossbar.
Dwight McNeil then fired into the side-netting and substitute Jay Rodriguez headed over in a late flurry for the Clarets.
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