The Coyotes took a 3-1 lead in the Western Conference semifinals and will try to close out the series at home Monday.
After failing to score against the Predators in Game 3 Wednesday, Phoenix managed only a first-period goal from Shane Doan in Friday's contest.
That was enough for Smith, who ranked among the top four goaltenders in victories and save percentage during the regular season. He improved his playoff record this season to 8-4 in his first stint as a front-line, post-season goaltender.
Smith's other playoff shutout this year came in the Coyotes' Game 6 win over Chicago that wrapped up the opening-round series.
Doan beat Pekka Rinne with a backhander 14:25 into the game.
This is the third straight year in the playoffs for Phoenix, who had sat out the post-season for seven consecutive seasons. The Coyotes are still looking for their first conference title.