San Jose's Logan Couture and New York's Kyle Okposo traded goals earlier in the shootout before Tavares beat Alex Stalock in the fifth round. San Jose's Joe Thornton missed on his turn and the Islanders improved to 4-0-0 before a raucous crowd at Nassau Coliseum.
The Islanders were perfect through four games previously to start in 1976-77 and 2001-02.
"These are fun games to play because there's such great energy in the building," said Tavares, who added two assists and leads the NHL with nine points. "We brought a good work ethic and a determination to win battles."
San Jose's Tomas Hertl tied the tense game at 11:16 of the third after goals by Okposo and Josh Bailey earlier in the period put the Islanders ahead 3-2.
Stalock made a spectacular glove stop on Tavares in the last minute of overtime and finished with 42 saves.
"The pace of the game was great, it was a real test of our players," Sharks coach Todd McClellan said. "But I thought they had more tenacity."
The Islanders dominated the first two periods, outshooting the Sharks 28-11 but they trailed 2-1 entering the third.
It was the first NHL point for the 19-year-old Mueller, a first-round draft pick by the Sharks in 2013. The goal was first of the season for Burns, who shrugged when asked if the wide shots differential between the teams was a true barometer of the game.
"Sometimes stats are weird," he said.
Nick Leddy tied it for the Islanders at 4:05 of the second, pouncing on a loose puck in the high slot and sending it past Stalock for his second goal.
Patrick Marleau put the Sharks back on top with a power-play goal at 9:14, scoring from the left point 7 seconds after Ryan Strome went off for cross-checking.
Marleau's team-best third goal this season was the 440th of his career. The 35-year-old forward is playing his 18th season with the Sharks.
The Islanders opened the season with two wins over the Carolina Hurricanes and a comeback 6-3 victory over the Rangers at Madison Square Garden on Tuesday. They next will visit the Pittsburgh Penguins on Saturday.
"Our puck management was good and we had good decision making," Islanders coach Jack Capuano said. "We did a lot of little things well."
San Jose's John Scott hammered Mikhail Grabovski midway through the first period with an apparently clean hit. Grabovski didn't return to the game. The Islanders said the forward sustained an upper body injury.
Earlier Thursday, the Islanders sent rookie defenseman Griffin Reinhart to AHL Bridgeport and activated Calvin de Haan from injured reserve.
NOTES: The Islanders wore the blue alternate jerseys they debuted for last January's outdoor game against the Rangers at Yankee Stadium ... The Islanders scratched Colin McDonald and Eric Boulton plus defenseman Matt Donovan. ... San Jose scratched Mike Brown and Mike Sheppard and defenseman Scott Hannan.