The Sharks started overtime with a 5-on-3 advantage thanks to penalties late in regulation handed out to Robin Regehr and Trevor Lewis.
Regehr had just gotten back on the ice when Patrick Marleau corralled Joe Thornton's feed behind the net and sent a touch-pass to Couture in the slot, who put a shot over the left arm of Jonathan Quick and into the Kings' net.
Dan Boyle also scored for the Sharks, who posted their first win in the conference semifinal series after dropping the first two games in Los Angeles.
Antti Niemi turned aside 26-of-27 shots for San Jose.
Tyler Toffoli scored and Quick finished with 38 stops for Los Angeles, which will try to grab a 3-1 lead in Tuesday's Game 4.