Nashville improved to 6-19-1 in franchise history in Dallas.
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.
Gabriel Bourque and Colin Wilson added even-strength goals for Nashville, 3-3-2 in its last eight after a five-game winning streak.
Kari Lehtonen made 18 saves for the Stars, 2-4-2 in their last eight.
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.
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.
NOTES: 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 and won the first two installments of the four-game season series by a total score of 10-4. ... Rinne was credited with the second assist on Wilson's goal.