The Norwegian’s first goal since his permanent departure from Real Madrid this summer proved decisive as Mikel Arteta’s side relied on last weekend’s home win over struggling Norwich.
Arteta’s side were indebted for a controversial intervention from VAR, which Burnley said deprived them of a penalty in the second half.
Referee Anthony Taylor initially indicated the spot when Burnley substitute Matej Vydra came under challenge from Arsenal goalkeeper Aaron Ramsdale, to change his mind after watching the replay.
Burnley led the opening 20 minutes in terms of half chances, with two efforts from forward Ashely Barnes sandwiching the lead from James Tarkowski.
Arsenal lacked the rhythm or creative spark to threaten Burnley’s well-established 4-4-2 system in the first 25 minutes, but that all changed after Barnes tripped Bukayo Saka in center position 25 yards from goal.
Barnes received a yellow card from referee Taylor and Odegaard brilliantly completed the resulting free kick beyond Burnley keeper Nick Pope.
It was a quality Premier League moment that the game was calling for and former Real Madrid midfielder Odegaard delivered his first goal of the season and third overall for the Gunners with left foot.
Burnley pushed forward in search of an equalizer before the break and Barnes, under pressure from Arsenal defender Gabriel, failed to shoot on target after Josh Brownhill’s ball was cut in the area.
Arsenal nearly doubled their lead at the stroke of half-time when Emile Smith Rowe completed a first shot just off the target after being selected by Nicolas Pepe.
Burnley’s intention after the second half resumed ignited Clarets fans and Ashley Westwood failed to get enough power on her effort at close range.
The Gunners responded with Smith Rowe, whose low shot was easily picked up by Pope after a good job from Saka and Pierre-Emerick Aubameyang.
Burnley boss Sean Dyche made a double substitution in the 56th minute, sending Maxwel Cornet and Vydra for Johann Berg Gudmundsson and Barnes, and it had an instant impact.
Cornet’s summer signing was greeted enthusiastically and the £ 12.85million buy from Lyon again uplifted the home crowd as it forced Ramsdale to make a quick stop a few minutes later.
Vydra also quickly contributed. He ran on Gunners defender Ben White’s short back pass and when the Czech Republic forward fell under Ramsdale’s challenge, referee Taylor awarded a penalty.
Ramsdale had contacted Vydra, but after VAR intervened, Taylor checked the replay and quickly reversed his decision.
The England keeper was on trial for touching the ball first and Taylor’s half-turn infuriated Burnley.
Chris Wood headed for the Clarets, Dwight McNeil shot into the side net and Vydra glanced at Charlie Taylor’s cross.
But Arsenal were keen to extend their league unbeaten streak at Turf Moor to 10 games and leave Burnley with a point in their first five games.
Additional reports by PA.