Vincent Kompany's team had briefly restored parity through Josh Brownhill after Leandro Trossard's courageous opener, but defensive duo William Saliba and Oleksandr Zinchenko propelled Arsenal to a routine victory, even if they did have to see out the final 10 minutes with 10 men.
There was good and bad news for Arsenal on the injury front prior to kickoff, as Bukayo Saka shook off an ankle issue to make the first XI, but Ben White was absent from the squad after sustaining a niggle against Sevilla in the Champions League.
While the Gunners dominated the ball early doors, David Raya had to be alert to beat away a Zeki Amdouni drive with just nine minutes gone, and James Trafford had little to do down the other end in the earliest exchanges.
That was until the 21st minute, when the Under-21 Euros winner produced a breathtaking one-handed save to prevent Saka's strike from finding the top corner, before the misfiring Kai Havertz sent a near-post header over the top on the half-hour mark.
Both number ones were coming up trumps as the first half progressed, with Raya denying Johann Berg Gudmundsson and Jay Rodriguez in quick succession straight after Havertz fluffed his lines, while Trossard was the next to be thwarted by an acrobatic Trafford save in the 36th minute.
However, Trossard succeeded with his second bite at the cherry as the clock ticked over into added time, nodding home at the back post after Saka headed Zinchenko's cross onto him, but the Belgian collided with the frame of the goal as he bravely committed.
Celebrations were somewhat muted as Trossard stayed down holding his forearm, and the Emirates contingent were fully silenced just nine minutes into the second half, as Luca Koleosho outmuscled Takehiro Tomiyasu on the right and cut back into the box; Arsenal failed to deal with the danger, and Brownhill found the net with a fortuitous deflected strike.
Arsenal felt that Tomiyasu had been illegally shoved over by Koleosho, but VAR saw nothing doing, although Burnley were only on level terms for three minutes, as Saliba got in front of Trafford to head Trossard's corner into the net from a yard out.
Another Trossard set-piece proved to be Burnley's downfall again in the 73rd minute, as Dara O'Shea inadvertently headed the Belgian's delivery onto his own crossbar, and the Clarets could only clear the ball as far as Zinchenko, who paid homage to Paolo Di Canio with a sumptuous scissor-kick into the top corner.
The hosts would not enjoy a simple cruise to the finish line, though, as substitute Vieira was given a straight red card for an awful studs-up challenge on Brownhill's knee, but Mikel Arteta's defence held out in a lengthy injury-time period to condemn the Clarets to another chastening defeat.
The Gunners' success moves them level on points with leaders Manchester City, who retain their first-placed position on goal difference, while Burnley remain in deep trouble in 19th spot.
Arsenal's first game back after the international break comes away to Brentford on November 25, while Burnley host West Ham United on the same afternoon.