Pavelski and Logan Couture each scored two goals, and Martin Jones made 23 saves for the fourth Stanley Cup Playoff shutout of his career, the previous three coming last season in San Jose's run to the Final.
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The Sharks tied their postseason record for most goals in a game and set their record with four power-play goals in eight opportunities.
Patrick Marleau, Marcus Sorensen and David Schlemko each scored, and Brent Burns had three assists for San Jose.
"That was something we wanted to do, obviously, was get one early," Couture said of Pavelski's goal, the fastest to start a playoff game in Sharks history. "This is an intimidating building to play in, and it is even more when you've got a lead early. [Pavelski] made a great tip on that one, the building got into it, and we rode that momentum."
Oilers goaltender Cam Talbot made saves on all 39 shots he faced in Games 2 and 3 but gave up five goals on 24 shots Tuesday and was replaced by Laurent Brossoit at 12:52 of the second period.
"We didn't start the night the way we wanted to," Talbot said. "I mean, we gave up a goal right away and we knew they were going to come hard after that and re-establish themselves. We talk about that 10-minute game and starting on the right foot, and we didn't do that tonight. We gave them all the momentum they needed to carry through the rest of the game."
Pavelski gave the Sharks the early lead when he redirected Justin Braun's shot past Talbot for his first goal of the playoffs. It was the first time San Jose opened the scoring in the series.