Streit got the puck at center ice and skated all the way in while stickhandling past a pair of Kings defenders, then put a backhander past Jonathan Quick 96 seconds into the extra frame for the game-winner.
Michael Grabner scored the Islanders' lone goal in regulation while Nabokov improved to 6-2-0 in his last eight starts.
Los Angeles, meanwhile, fell for the fourth time in five contests as Quick finished with 23 saves.
Andrei Loktionov tallied the Kings' only goal.
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