The Islanders were whistled for a penalty just 29 seconds into the extra frame when Travis Hamonic shot the puck over the glass and Burns capitalized on the 4-on-3 power play 39 seconds later.
Burns also assisted on Logan Couture's score-tying tally in the second period. Joe Pavleski added a goal and an assist for the Sharks, who improved to 5-0-0 on their six-game road trip.
Thomas Greiss turned aside 35-of-37 shots for San Jose.
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