Nashville improved to 6-19-1 in franchise history in Dallas.
"It's been a tough building," Predators coach Barry Trotz said.
Weber put the Predators ahead to stay 2-1 with his 12th goal of the season, popping a loose puck into an open net with 0.2 seconds left in the first. The goal was upheld after a video review showed the puck entered the net before time expired.
"I saw it go in, I heard the horn ... so I assumed it was in, but I didn't realize how close it was," Weber said.
The Stars won the season's first two meetings against the Predators by a total score of 10-4. This time, the Predators dictated the kind of close-checking style they favor.
"We haven't played well against Dallas at all this year so it was good to get the win," Weber said.
"Another one-goal game," Goligoski said. "I think for the most part, we were maybe outworked a little bit tonight. The difference is that we give them maybe a freebie there at the end of the first. ... We were working hard, just not hard enough."
Kari Lehtonen made 18 saves for the Stars, 2-4-2 in their last eight.
Dallas coach Glen Gulutzan thought his team played well in stretches, but the Stars continue to slump as they try to climb into the top eight in the West.
"As hard as it is to swallow, you've got to keep doing things right," Gulutzan said. "The moment you veer away and start losing confidence, you're in trouble. We're doing the little things right, we're not getting the result."
First-line center Jamie Benn, Dallas' only All-Star pick, missed his second straight game with a knee injury and is day-to-day.
The Stars turned up the pressure in the third period, but Rinne made a sprawling save on Tomas Vincour's close-range attempt with 17:22 left, then denied Eric Nystrom's choice scoring chance with 15:30 remaining.
Goligoski got the Stars within one at 15:11 of the third period when his shot from the edge of the left circle sneaked past Rinne for Goligoski's eighth of the season.
Dallas went on the power play with 2:05 left when Nashville's Hal Gill went to the penalty box for tripping. The Stars pulled Lehtonen with 1:25 remaining for a 6-on-4 advantage, but couldn't come up with the equalizer.
Gill, acquired in a trade with Montreal on Friday and making his debut for the Predators, logged almost 17 minutes.
"I don't think there's a secret what my game is, what my strengths are: kill penalties, move the puck quick, use my reach and try to control the front of the net," Gill said.
Mike Fisher, Nashville's top goal-scorer with 17, was scratched due to illness.
Ryder gave the Stars the lead at 6:49 of the opening period, notching his team-best 23rd goal from the slot off Mike Ribeiro's centering pass.
Bourque's re-direct off David Legwand's setup for his third of the season drew the Predators even at 1 at 15:39 of the first.
After Weber gave them the lead, the Predators thought they'd made it 3-1 midway through the second period when Matt Halischuk knocked in a rebound. But the goal was disallowed when a video review showed that Halischuk directed the puck into the net with his glove.
Wilson officially extended Nashville edge to 3-1 at 16:05 of the second period on a backhander from the low slot, Wilson's 12th of the season.
- The last time Weber played in Dallas on Dec. 23, he sustained a head injury on a hard hit by Stars defenseman Mark Fistric, forcing Weber to miss four games.
- LW Brenden Morrow (neck, shoulder) and D Sheldon Souray (foot) remain out for Dallas.
- In addition to Fisher, the Predators were missing forwards Patric Hornqvist (upper body) and Brian McGrattan (upper body).
- The Stars were 5-1-1 in their last seven at home against Nashville.
- Rinne was credited with the second assist on Wilson's goal.