Hedman had six shots on goal for Tampa Bay (38-11-2), which has scored two or fewer goals in five of its past six games. Steven Stamkos' seven-game point streak ended.
Vasilevskiy made four saves in overtime, including two during a penalty kill after Nikita Kucherov cross-checked Nick Leddy behind the Islanders net with 1:37 remaining.
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"We played good defensively, and they didn't have a lot in the first 10, 15 minutes," Vasilevskiy said. "It helps a lot."
Thomas Greiss made 41 saves for New York (29-15-6), which extended its point streak to seven games (5-0-2). The Islanders are 15-3-2 in their past 20 games.
Greiss, who made 16 saves in the third period, has allowed one goal in his past three starts. It was his third shutout of the season and second in his past two games.
"Both goaltenders were outstanding, they made some huge saves," Islanders coach Barry Trotz said. "There was some physicality in the game, there was some excitement in terms of good defensive plays and good offensive plays … a little bit of scrambly play, which happens, and the intensity on both sides."