Arsene Wenger's side appeared to have blown their chance to get back in the title race when Burnley's Andre Gray scored a stoppage-time penalty to cancel out Shkodran Mustafi's first goal for Arsenal.
Xhaka tips Arsenal to close in on leaders Chelsea
Arsenal had been forced to play the last 25 minutes with 10 men after Granit Xhaka was shown a straight red card for a reckless lunge on Steven Defour.
But the Gunners refused to be denied and Sanchez converted from the spot with virtually the last kick of an incredible finish at the Emirates Stadium to ensure his team leapfrogged Tottenham and Liverpool in the title chase.
Ozil and Arsenal look to bounce back after mini-slump
The north Londoners have now won four of their last five league games and remain in touch with leaders Chelsea ahead of their summit meeting at Stamford Bridge on February 4.
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