Justin Williams had two goals for the Kings, who began the night one point behind San Jose for the final playoff spot in the Western Conference.
Los Angeles took a one-goal lead on three occasions only to have the Predators jump back into a tie.
Greene, however, put the Kings in front to stay with a slap shot that eluded Pekka Rinne early in the third. Trevor Lewis added an insurance goal for Los Angeles and that score became the game-winner when Patric Hornqvist tallied for Nashville with 5:18 to go in regulation.
The goal by Lewis drove Rinne from the net and he was replaced down the stretch by Anders Lindback.
The win went to Jonathan Quick, who made 22 saves.