Nashville got its only power play of the game with 4:46 remaining in the third when goaltender Jimmy Howard was assessed a hooking penalty on Tootoo. Only 22 seconds after the penalty was called, Weber's blast from above the left faceoff circle beat a screened Howard on the glove side.
The Predators have scored a power-play goal in seven consecutive games.
Weber's wrist shot from the right point just under three minutes later bounced in front of the net and slipped by Howard for the eventual winner.
Holmstrom scored the game's first goal at 5:15 of the opening period.
With the Red Wings on a power play, White's shot from the right point deflected off two Predators before landing on the blade of Holmstrom's stick just to the left of Nashville goaltender Pekka Rinne. Holmstrom had an easy tap-in for his sixth goal of the season.
Legwand drew the Predators even midway through the first when he collected his own rebound to the right of the Detroit goal and put it past a diving Howard.
Detroit regained the lead with 9:57 to go in the second on Zetterberg's power-play goal.
Prior to Thursday night, Nashville had not allowed a power-play goal in its last five games.
Lidstrom gave Detroit a 3-1 lead with 8:30 left in the second when his slap shot from the left point beat Rinne high to the glove side.
Nashville answered with 2:38 remaining in the period on a goal by Tootoo, who converted the rebound of Kevin Klein's shot from the left side.
Tootoo has nine points in his last ten games.