The Sharks clawed back into the best of seven series after Couture took advantage of a power play penalty just 1:29 into the extra period.
The Kings' Robyn Regehr (hooking) and Trevor Lewis (interference) were both penalized inside the final minute of regulation to give San Jose a five-on-three situation into overtime.
While they failed to score with a two-man advantage, Couture rammed home a chance while still in the five-on-four powerplay from a Patrick Marleau pass to send the home crowd into rapture as they dragged themselves back into the series.
Dan Boyle had given the Sharks a dream start with a goal after just 94 seconds but 21-year-old Tyler Toffoli cancelled it out with a goal midway through the first period for the Kings.
"You don't want to go down three games, this was a big game. It's a known fact we needed a win," Boyle said of the result that left the series at 2-1 to the Kings with Game Four also in San Jose on May 21.
(Reporting by Ben Everill in Los Angeles; Editing by Greg Stutchbury)
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