With the game tied, 1-1, and Vancouver's Matt Cooke in the penalty box, Jillson took a cross-ice pass inside the blue line from Vincent Damphousse and fired a shot through a screen that beat goaltender Martin Brochu 3:03 into the period.
Eight seconds later, defenseman Mattias Ohlund coughed up the puck in the Sharks' zone. Thornton moved into the Canucks' end and put a shot between Brochu's pads as he went down to make the save.
Sturm made it 4-1 on the power play 28 seconds thereafter. Teemu Selanne passed from behind the net to the top of the crease, where Sturm poked the puck past Brochu with Vancouver right wing Trent Klatt behind him.
San Jose broke the team record for quickest three goals. Sergei Makarov, Ray Whitney and Gaetan Duchesne scored in a 38-second span on March 29, 1994 against the Winnipeg Jets.
Cooke gave Vancouver the early lead 3:07 into the contest when he carried over the blue line and fired a shot past goaltender Evgeni Nabokov for his second goal of the season.
Nabokov stopped 33 shots for his fifth win in six games, improving to 6-3-2 this season.
Swim Week Miami Beach 2014 [PHOTOS]