For the second year in a row the fixture between the two was decided in controversial circumstances as Arsene Wenger's men overcame the Clarets thanks to some late drama.
Last season Laurent Koscielny scored a contentious winner as the ball appeared to strike his hand late on, but this year it was a penalty that will leave Burnley manager Sean Dyche disappointed.
Gunners struggle away from home again
As expected, Arsenal found life tough at Turf Moor and Petr Cech had to be at his best to keep things level in the first half.
Numerous balls were lifted into the area, but it was from other positions where Cech was forced to spring to life.
Johann Berg Gudmundsson got past his man after 13 minutes played and then smacked a shot which the Arsenal keeper tipped onto the post.
Robbie Brady was next to fire a shot at the visitors goal. Koscielny brought down Ashley Barnes, and from the resulting set piece saw his effort denied.
Arsenal's best chance of the first half came after 19 minutes. Alexandre Lacazette burst down the left channel before picking out Aaron Ramsey in the middle.
The Welshman was unmarked but couldn't find the net when he really should have buried the chance.
Burnley restrict Arsenal in attack
For the most part the Gunners had to make do with efforts from outside the box.
Sanchez drove wide in the opening minutes and Nacho Monreal twice fired off target. A Granit Xhaka attempt from distance meanwhile flew over as the team's went into the break all square.
The second half brought of the same, as the Gunners continued to struggle against a resurgent Burnley side.
Arsenal enjoyed the better spells but couldn't fashion a clear cut opportunity with a team missing Mesut Ozil through illness.
Sead Kolasinac had a shot blocked by Ben Mee from inside the area shortly after the break, and then two chances went begging for Lacazette.
He first saw a tame effort from outside the box easily gathered by Nick Pope before an effort just after the hour mark was deflected wide for a corner.
Wenger brought on Jack Wilshere and Danny Welbeck to change the game, and the former almost opened the scoring. He linked up well with Sanchez with ten minutes remaining but Wilshere's effort hit a Burnley man and bounced wide.
Sanchez wins it late on
With the game seemingly heading for a 0-0 draw, Arsenal won a late penalty. James Tarkowski tangled with Ramsey and controversially referee Lee Mason pointed to the spot.
It was Sanchez who stepped up, and he beat Pope despite the Burnley keeper diving the right way as the Chilean gave the Gunners a late win at Turf Moor again.