Ivar Stenberg scored in regulation and had the only goal in the shootout. He skated down the left wing before cutting toward the middle and beating New York goalie Ilya Sorokin over his glove with a wrist shot.
Tyler Toffoli had a goal and an assist for the Sharks (8-11-1), who won for just the second time in their past nine games (2-7-0).
Victor Eklund and Ben Hutton scored for the Islanders (7-6-4), who have lost four straight (0-2-2). Sorokin made 37 saves.
Hutton put the Islanders up 1-0 at 6:14 of the first period. Simon Holmstrom was at the right point and sent a diagonal pass down to Hutton, who had chipped down and beat Askarov's right pad at the back post.
Stenberg tied it 1-1 at 16:41. Ty Dellandrea knocked down Logan Stanley's clearing attempt up the boards, and Toffoli's wrist shot from the top of the right circle deflected in off the stick of Stenberg.
Eklund put the Islanders back in front 2-1 with a power-play goal at 1:11 of the second period. He scored with one-timer that beat Askarov over his glove after Brayden Schenn fed a short pass to the right hash marks.
Toffoli tied it 2-2 at 12:12, scoring six seconds after a power play expired. Dmitry Orlov's initial shot kicked off Sorokin's right pad into the left circle, where Toffoli scored with a shot that deflected in off Islanders defenseman Adam Pelech's face.
