Neither side could score in the shootout until Kopitar beat Pekka Rinne to bring the Kings their fifth victory in eight games this season.
Matt Cullen gave the Predators the lead midway through the first and Los Angeles tied it with 42 seconds to go in the opening stanza on a goal by Shane Voynov.
Rinne stopped 29-of-30 shots in regulation and overtime before giving up the game-winner to Kopitar.
Jonathan Quick was credited with 34 saves in the victory.
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