Mikkel Boedker also scored for the Coyotes (10-3-2), who have points in 11 of their last 12 games.
Joe Pavelski and Danny Boyle, who missed the previous seven games with a concussion, scored for the Sharks (10-1-3).
Logan Couture converted the first attempt of the shootout for San Jose. He earlier thought he had the winning goal with 33 seconds left in regulation. However, it was waved off and replays clearly showed the puck never crossed the line.
It was just one of many brilliant saves made by Mike Smith, who stopped 48 shots through overtime. Sharks goalie Antti Niemi made 28 saves.
Radim Vrbata also scored in the shootout for Phoenix.
Phoenix peppered Niemi with seven shots during the 5-minute overtime, with Niemi coming up big. The Sharks had some chances but did not get a shot on goal.
The Coyotes scored just 36 seconds into the game. Rob Klinkhammer picked up a loose puck along the boards near the blue line and fed Vermette, who beat Niemi to the far post.
Pavelski's goal came just over a minute into the second period, tying the score at 1. Tommy Wingels won a scrum in the Sharks' zone and sent it up the ice to Matthew Nieto, who missed the previous two games with an injury.
Pavelski took the pass and skated in, looked off Smith and found an opening to the near side.
The Sharks took a 2-1 lead with a power-play goal four minutes later. Pavelski's shot was blocked and the Coyotes couldn't clear it and Thornton kicked it back out to Pavelski, who found Boyle on the left side. Boyle put it into the upper far corner.
Boedker took a pass from Derek Morris midway through the second period and slipped it along the ice to the far side to tie the score at 2.
NOTES: Sharks C Joe Thornton recorded his 800th career assist on Boyle's goal. ... Coyotes C Mike Ribeiro played in his 800th career game. ... Boedker ended a three-game scoreless streak. ... Vermette has scored goals in consecutive games. He did not record a point in his previous four. ... Morris has recorded points in eight of his last 11 games. ... The Coyotes are outscoring their opponents 22-15 in the second period. ... Pavelski has scored goals in three straight. ... The Sharks have taken 50 or more shots in both home games against the Coyotes this year. ... Vrbata improved to 32 of 73 on penalty shots.