Pierre-Emerick Aubameyang, on target in last weekend's win at Newcastle United, fired in his second decisive goal in as many games after the hour to move Unai Emery's men to six points from two matches.
Alexandre Lacazette earlier opened the scoring in the 13th minute, only for Ashley Barnes to respond in fortuitous fashion shortly before the break.
An impressive performance from Dani Ceballos inspired Arsenal's recovery and debutant David Luiz held the home side's defence together to give the Gunners an 11th straight win over Burnley in all competitions.
Lacazette tested goalkeeper Nick Pope with a close-range header before scoring from the resulting corner.
With his back to goal at the near post, the France forward managed to spin and squeeze a shot through Pope's legs.
Pope remained busy until Barnes equalised two minutes from half-time, cushioning Dwight McNeil's deflected effort and prodding home despite Matteo Guendouzi's best efforts.
Reiss Nelson had the ball in the net a few moments later, but Nacho Monreal was correctly adjudged offside in the build-up.
That proved winger Nelson's final act as club-record signing Nicolas Pepe entered the fray at half-time.
Pope produced another two fine saves either side of the hour but could do little about the 64th-minute winner.
Ceballos was the architect of the move as he won the ball back in midfield and fed it forward to Aubameyang, who rifled into the bottom left corner to end Burnley's resistance.
Arsenal travels next to Merseyside for a difficult date with Liverpool next weekend, while Burnley visits Wolverhampton Wanderers.