The Isles got the 4-on-3 advantage when Brett Lebda was whistled for interference. P.A. Parenteau passed back to the point for James Wisniewski, who found Tavares for a one-timer that beat Jean-Sebastien Giguere on the blocker side.
Matt Moulson notched New York's other goal early in the second period and Dwayne Roloson turned aside 29-of-30 shots in the victory.
Phil Kessel collected his fifth goal for the Leafs, who lost for the first time after four straight wins to start the season.
Giguere stopped 18 of 20 Islanders shots.
Interpol investigating stolen passports on missing flight
Scarlett Johansson steps out with fiance after pregnancy reveal