Jan Hlavac tipped a pass to Peverley, who fired a shot from the slot past Hannu Toivonen to win the game 1:43 into the extra session.
Dan Ellis, who had allowed just one goal in more than 233 minutes, was pulled after allowing three goals on seven shots. Chris Mason stopped all 10 shots he faced the rest of the way and earned the win.
The win temporarily bumped the Predators into the eighth and final playoff spot in the Western Conference ahead of the Vancouver Canucks.
2014: The Year in Fashion [PHOTOS]
LeBron James reveals his new Nike shoes