Slava Voynov scored for the Kings, who have won four of five.
Kopitar skated down the right side and elevated a backhand over the left shoulder of Nashville goaltender Pekka Rinne.
Matt Cullen scored his first goal with the Predators at 10:33 of the opening period for the early lead. Nashville finished a five-game homestand 3-2.
After Kings goaltender Jonathan Quick made a diving save to deny Nashville's Gabriel Bourque from the slot, Cullen was able to corral the puck in traffic, move to open space, and then beat a fallen Quick with a wrist shot from the right side.
Nashville has scored first in three consecutive games.
Los Angeles tied it just before the first intermission when Voynov scored at 19:18.
At the left faceoff dot, Mike Richards froze Rinne and then fired a cross-ice pass to Voynov in the right circle, where he was able to beat a diving Rinne for his second goal of the season.
Voynov has scored in consecutive games.
With just over six minutes remaining in the third, Rinne denied Mike Richards on a short-handed breakaway attempt to keep the game tied at 1.
Nashville had the only power-play opportunity of overtime, but Quick denied both Nashville shots on the man advantage, both by Patric Hornqvist.
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