The Predators (27-9-4) moved into first place overall in the NHL with 58 points and are 16-2-1 at home.
"It's still a long season, but it definitely means you've done some good things this year," Predators defenseman Roman Josi said. "I think we've been playing really consistent. We've got to find a way to keep it going like that in the second half."
Weber won the game 17 seconds into overtime on a wrist shot that deflected off Dallas goaltender Kari Lehtonen and rolled into the net. The goal was Weber's 10th of the season.
"We knew it was going to be a challenge," Weber said. "They made a push. Another good job of coming back after a disallowed goal and sticking with our game."
Weber has four goals in the past five games after not scoring for 16 games. He is third on the Predators in scoring with 10 goals and 28 points.
"You know if he doesn’t score for a while, he's going to score," Josi said. "His shot is the best shot in the League. Not just his shot, his presence on the ice. He's our leader. He leads by example on the ice, you saw it again. When it's important, he steps up and scores the game-winner tonight. You can't describe how important he is for our team."
Weber gave the Predators a 1-0 lead at 15:27 of the first period on a one-timer from the point. Forward Mike Ribeiro delivered a drop pass that Weber was able to slap past Lehtonen. Josi’s assist on the goal was his 100th NHL point.
"You guys need to stop talking about [Josi] so people don't worry about him," Weber said. "He's been so great for us. Our whole defense corps, our young defense have [taken another] step this year. He's just one of them that is playing at another level right now, so hopefully he keeps that up."
Dallas (18-15-7) tied the game on a goal by Jason Spezza at 12:51 of the second period on a wrist shot from a tough angle. Spezza saw some space between Nashville goaltender Pekka Rinne and the post and was able to shoot the puck past Rinne. Forward Shawn Horcoff had the primary assist in his return to the lineup after missing four games because of illness.
"They defend really well," Spezza said. "Their defense is one of the best in the League. They have a tight gap. They outplayed us in the first, I thought we outplayed them in the second. It was a pretty even game, and then it comes down to overtime and special teams."
Nashville forward Miikka Salomaki scored his first NHL goal in his first game to give the Predators a 2-1 lead at 16:47 of the second period. Salomaki’s breakaway attempt was stopped by Lehtonen, but he was able to collect the rebound and score.
"I have no words," Salomaki said. "It was just an awesome feeling. I think you can imagine it. It has been a dream for me to play in the NHL and to have my first game and first goal is so fun."
Dallas defenseman Trevor Daley tied the game 2-2 with 1.6 seconds remaining in the second period. Daley took the pass from forward Jamie Benn and was able to jam the puck past Rinne from point-blank range.
"I felt like we did some really nice things out there," Lehtonen said. "We've had success here before. Everybody knows they've been playing well, but we're also confident when we come here. It was a tight game today. We just have to learn to turn them the other way."
Forsberg appeared to give Nashville a 3-2 lead with 3:47 left in the third but forward Craig Smith made incidental contact with Lehtonen. The goal was disallowed after review, which was Forsberg’s second disallowed goal of the season.
"I haven't really seen the replay or anything, but everyone has been telling me it's a goal," Forsberg said. "Obviously it's frustrating right there, but it's good that we got the goal back later on."
Nashville defenseman Ryan Ellis left the game in the second period with a lower-body injury and did not return. Forwards James Neal and Eric Nystrom were placed on injured reserve Thursday in addition to forwards Taylor Beck and Matt Cullen.
"You don't want to lose key guys but when you do and guys step up, it's a big boost for your club," Weber said. "That's what you need. An 80-game season is a long year, and things are going to happen."