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.
Celebrity Families of 2014 [PHOTOS]