The Nashville defenseman scored his second goal of the game on a power play in overtime to give the Predators a 6-5 victory Thursday night.
"Definitely not a defensive battle. It was a lot of offense, but we got two points and we're happy," said Weber, who also had an assist.
Weber ripped a shot between goalie Pascal Leclaire's pads with Matt Carkner serving an interference penalty. The goal put an end to a game that saw the Predators blow a 3-0 lead in the third period and the team's exchange last-minute goals.
Chris Phillips scored the tying goal for the Senators with 1.6 seconds left while Ottawa was short-handed and Leclaire on the bench for an extra attacker.
"That's a tough game to sum up in 30 seconds. A lot happened," Phillips said. "Obviously, not the start we wanted, but we battled back and lots happened in between there, a lot of goals scored."
Rookie Cody Franson also scored for Nashville. It was the first career goal for Franson, who was playing in just his third NHL game.
Dave Scatchard, a veteran of 637 NHL games who was recalled Tuesday from Milwaukee of the AHL, recorded an assist, his first NHL point since Jan. 7, 2007. Rookie Alexander Sulzer, also recalled Tuesday, had his first career point with an assist.
"We've had some of the craziest games here in this rink," Nashville coach Barry Trotz said. "They scored, we scored, the third period was like that. We had a 3-0 lead and we're in good shape, then we took penalties. But give our guys a lot of credit because we've been a little bit fragile lately.
"You're on the power play and they score a short-handed goal with 1.6 seconds left and you're on a six-game losing streak, that plays with your mind."
Pekka Rinne made 27 saves for his first victory of the season.
Anton Volchenkov had a goal and an assist, and Chris Campoli, Nick Foligno and Milan Michalek also scored for Ottawa. Jason Spezza had three assists, Daniel Alfredsson added two, and Leclaire finished with 31 stops.
"I think we showed character in the way we fought back," Alfredsson said. "It's a little frustrating, but the way we found a way to get a point is very encouraging."
Franson's goal marked the second straight night a member of the Predators has scored his first NHL goal. Colin Wilson had his first Wednesday against Boston. ... Nashville is at Chicago on Saturday night.