Stamkos made it 4-all, scoring from the slot 2:44 into the third.
St. Louis gave the Lightning a 5-4 lead at 8:01 when his shot from along the goal line went off San Jose's Justin Braun and past goalie Antti Niemi.
"It went off my shaft, took a bad bounce and ends up in the net," Braun said. "It happens. I probably shouldn't be that close to the blue paint, but you have to learn from it."
Tampa Bay's Dwayne Roloson made 45 saves, including five in overtime. Niemi stopped 19 shots.
"He wasn't any good," San Jose coach Todd McLellan said about Niemi. "I protect him a lot and talk about the team. He has to be better, flat out has to be better. They scored four goals on net play, rebounds, picking up the garbage, wraparounds, jams. That's our D-men. That's our low forwards. So everybody has some responsibility, but [Niemi] knows he has to be better."
Vlasic got his first goal in 17 games at 14:37 of second, and White stopped a personal 148-game goal drought (dating to Dec. 28, 2009) 67 seconds later as San Jose went ahead 4-3.
Tampa Bay, outshot 20-4 during the second, had taken a 3-2 advantage on Downie's goal at 14:08.
Handzus tied it 2 from the low slot at 1:44 of the second.
After Stamkos scored at 9:30 of the first, he assisted on Purcell's power-play goal that gave Tampa Bay a 2-1 lead with 24.4 seconds left in the first.
Couture put the Sharks up 1-0 on a power-play goal just 1:16 into the game. The center had two goals in San Jose's 7-2 victory over Tampa Bay on Dec. 21.
San Jose D Douglas Murray left with an injury after playing just two shifts. ... Lightning D Brendan Mikkelson snapped a 31-game point drought, dating to Oct. 30, 2010, with two assists. He has no goals and eight assists in 103 NHL games. ... Sharks C Torrey Mitchell had two assists for his first points in 12 games. ... Tampa Bay D Matt Gilroy was back in the lineup after missing two games because of a hand injury.
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