Burnley looked like taking their second point on the road this season when substitute Francis Coquelin tripped Ashley Barnes and Andre Gray powered home the first spot-kick in stoppage time to cancel out Shkodran Mustafi's header.
But, in the last of the seven additional minutes, referee Jon Moss - who had already dismissed Granit Xhaka and sent Arsene Wenger to the stands - pointed to the spot once more after Ben Mee's high foot connected with Laurent Koscielny.
Wenger had been infuriated by the decision to award the first penalty. The Arsenal manager was sent off and stopped near the tunnel to watch the rest of the match and Sanchez's winner, before getting into a heated discussion with Anthony Taylor and appearing to push the fourth official away.
Wenger could face action from the English Football Association.
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"I regret everything," Wenger said. "I should have shut up, gone in and gone home. I apologize for that."
"I said something that you hear every day in football," Wenger recalled. "Overall nine times out of 10 you are not sent to the stand for that. If I am, I am and I should have shut up completely. I was quite calm the whole game, more than usual."
Wenger also insisted that Xhaka needs to show greater control. Xhaka, who joined last year from Borussia Moenchengladbach, was shown his ninth red card since 2014 - and second for the Gunners - after a two-footed lunge on Steven Defour in the 66th minute.