The Clarets haven't won at home since the end of January when they came from behind to beat Aston Villa.
Sean Dyche's side faced an uphill task in arresting that run when Martin Odegaard gave the Gunners the lead on the half-hour mark.
Arsenal, who hadn't experienced a league defeat at Turf Moor since 1973, hadn't registered a shot on target up until that point.
But Nick Pope's unnecessarily rushed clearance, in front of England boss Gareth Southgate, gifted possession back to the visitors.
Mikel Arteta's men came forward through Bukayo Saka, who was clipped on the edge of the box by Ashley Westwood.
Odegaard, 22, who made his move to the Emirates permanent for £30m in the summer, stood over the ball alongside two team-mates.
The Norwegian swept the ball elegantly over the wall and beyond the fingertips of Pope to break the deadlock.
Burnley had offered little before the break, with Ashley Barnes wasteful in front of goal, but they improved in the second half.
The introduction of debutant Maxwel Cornet alongside Matej Vydra before the hour gave the hosts the lift they had craved.
Aaron Ramsdale denied Matt Lowton after Johann Berg Gudmundsson's attempt was blocked when arrowing towards the corner.
Burnley thought they would have the opportunity to level in the 68th mi9nute when referee Anthony Taylor pointed to the spot once Vydra had tumbled under Ramsdale's challenge.
Chris Wood walked the ball towards the spot as the visitors protested, with the official marching over to the Bob Lord Stand to check his monitor.
After a quick check, Taylor decided that the ex-Sheffield United stopper had got a toe to the ball and the decision was overturned.
Moments after, Cornet almost marked his debut with a goal when the ball broke kindly for the Ivorian inside the box.
Ramsdale, though, hadn't read the script and the England Under 21 international turned his shot over the bar.
The home side applied the pressure, but Ramsdale remained largely untested.
In a late bid for an equaliser, Dwight McNeil fired into the side-netting, with some fans on their feet, before substitute Jay Rodriguez headed wide.
