Anaheim appeared headed for a win in regulation but Shea Weber scored for Nashville with 36 seconds left in the third period to force overtime.
Smithson then quickly ended the game in the first extra period by one-timing a shot past Ray Emery off a feed from Jordin Tootoo. Smithson played in all 82 games in the regular season and scored just six goals.
The Predators took a 3-2 lead in the Western Conference quarterfinals and will try to move into the second round of the playoffs when they host Game 6 Sunday.
Kevin Klein and Joel Ward also scored for the Predators, who won for the second time in Anaheim.
Jason Blake scored twice for the Ducks, producing the only goal of the second period to tie the score 1-1 and then giving Anaheim a 3-2 lead with 5:44 to go in the third.
Pekka Rinne stopped 23 shots to get the win.
Emery allowed four goals on 37 shots.