Rinne improved to an NHL-best 35-13-7 overall and 16-2-3 since Jan. 7 with the performance, the Predators' fourth win in five games.
Legwand helped out on Roman Josi's first-period goal and on Patric Hornqvist's 18th of the season in the middle frame for Nashville.
The Kings scored with 3 1/2 minutes to go in the game when Dustin Brown got credit for a goal that bounced off several players before getting past Rinne, but that was all the closer Los Angeles would get.
Jonathan Quick absorbed the hard-luck loss, stopping 26-of-28 shots as the Kings remained two points out of the eighth and final playoff spot in the Western Conference.
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