Goligoski delivered a lucky goal, then set up the game-winner to give his new team two precious points in a tight playoff race.
Earlier in the third period, Goligoski scored his first goal since being acquired by the Stars from Pittsburgh on Monday. He tied it at 2 at 6:02 when his dump-in hit defenseman Jonathon Blum, changed direction and fluttered by Rinne.
"You could see where Goligoski's speed and ability to join the rush just adds creativity to your offensive game, when you can create an outnumbering chance, and he knows what to do with the puck," Stars coach Marc Crawford said. "He made a good pass to initially start it, he followed the play, and then made a quality pass to Loui Eriksson to put it home."
Dallas has won two straight for the first time since Jan. 17-20, beating one of the teams the Stars are fighting for a playoff spot and stopping a slide that knocked them out of first place in the Pacific Division.
Now the Stars face a four-game road trip, all against playoff contenders, starting Tuesday at Phoenix.
"Trailing going into the third period, a big road trip coming up, this was a big game, and obviously where (the Predators) were in the standings and where we were, it was a big win, definitely," Goligoski said.
Weber refrained from complaining about the penalty call for his hit against Dallas rookie Tomas Vincour.
"I'm not going to say anything because I'll regret it," Weber said.
His teammates, however, said more in voicing their displeasure with the call.
"That's been in the game since it was invented," said Predators forward David Legwand, who had two assists. "I didn't know you weren't allowed to hit out there."
"It's nice to score, but it hurts to lose just as much," Weber said.
With the Predators controlling play early in the final period, Goligoski's goal gave the Stars momentum.
"They got a fluke goal, plain and simple," Nashville coach Barry Trotz said. "We defended well and did all the things you have to do in the third period. They were just dumping it in. Please. That's a one-in-a-million thing."
Just 12 seconds after Dallas killed Nashville's first power play, Wandell finished off a 2-on-1 break by firing a shot from the right circle that got between Rinne's pads with 1:54 remaining in the opening period.
Nashville tied it at 5:00 of the second when Suter sent a loose puck past Lehtonen from the slot. Wilson's shot bounced off the skate of Stars defenseman Stephane Robidas and onto Suter's stick.
Wilson gave the Predators a 2-1 lead by shooting a rebound of Cody Franson's drive past Lehtonen 13:32 into the second.
In Nashville's only other visit to Dallas this season, Rinne made 30 saves in a 1-0 win on Oct. 23.
- The Predators are thin at center with Marcel Goc (shoulder), Matthew Lombardi (concussion) and Cal O'Reilly (broken fibula) all out for the season.
- Stars C Brad Richards (concussion-like symptoms) missed his sixth straight game. Richards has been skating at practice, but isn't ready to play. Richards, who can be an unrestricted free agent this summer, continues to be the subject of trade rumors as Monday's NHL trade deadline approaches.
- D Nicklas Grossman (hip) sat out his sixth consecutive game.
- The Stars are 18-5-1 against the Predators in Dallas.