Burnley looked destined to claim a point at Arsenal after Andre Gray had fired in from the spot in stoppage time.
But Arsenal were awarded a penalty with seconds remaining when Ben Mee was penalised for a high boot on Laurent Koscielny.
Alexis Sanchez stepped up to take the spot kick and confidently pulled off a Panenka-style chip to beat Tom Heaton.
Alexis Sanchez scored from the spot with seconds remaining in the game
Arsene Wenger labeled Alexis Sanchez "too cool" for his chipped penalty
Arsene Wenger was watching from the tunnel at this point after being sent off but admitted the Chile international may not have even taken the penalty had Danny Welbeck not replayed Olivier Giroud.
"He was too cool," Wenger said.
"He has missed penalties for us, he was in fact, not the first penalty taker today – it was Giroud.
Arsenal 2-1 Burnley: 10 key stats
Sun, January 22, 2017
Express Sport brings you 10 defining statistics from Arsenal's victory over Burnley at the Emirates
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Express Sport brings you 10 key stats from Arsenal's win over Burnley
Olivier Giroud would have taken the penalty had he been on the field
"But he was second. But as long as he scores, you know, we just have to accept and say 'well done.'"
The Arsenal boss is facing a touchline ban for his furious outburst at fourth official Anthony Taylor after Burnley's equaliser.
Wenger was apologetic for his behaviour but is expected to face retrospective action from the FA.