It marked the third straight game Arsene Wenger’s side have scored an injury time winner against the Clarets. Laurent Koscielny’s scored a controversial winner at Turf Moor in September 2016, while Sanchez tucked home a 98th minute penalty to give the 10-man Gunners a 2-1 victory in January.
The result was harsh on Burnley, who had been on top for large amounts of the game.
Referee Lee Mason pointed to the spot after defender James Tarkowski appeared to push Aaron Ramsey in the back.
Clarets boss Sean Dyche struggled to keep his cool in his post-match interview.
“It is a coupon buster,” said Dyche (via BBC Sport). “I don’t think anyone would put a bet on that.
“I have told the players we continue to develop, we continue to move forward. I have also told them this is the reality. Football is a harsh game and sometimes it hurts you.
“There is a lot of me inside that is raging a lot. It was highly unlikely it was not going to get given on a day like today, that’s all I will say.”
Wenger’s side have moved back into the top four, but remain 12 points behind runaway leaders Manchester City.
Arsenal are now preparing for their clash with Huddersfield on Wednesday night.