Legwand scored his second goal of the game 2:53 into overtime to lift the Predators to a 4-3 victory over the San Jose Sharks on Saturday night.
Rinne, who signed a 7-year, $49 million deal on Thursday, made several key saves down the stretch for the Predators, who overcame a two-goal deficit in the third period. He finished with 35 saves.
The two were able to celebrate together with their traditional victory song: Lionel Ritchie's "All Night Long."
"I don't think they know who Lionel Ritchie is," Nashville coach Barry Trotz said. "That's my genre."
"It was a big comeback," Trotz said. "They put a lot of pucks on the net in the first period but we stayed resilient, and Leggy - deservedly so - gets the winner."
Joe Pavelski, Martin Havlat and Logan Couture scored for the Sharks, who lost for the second time in three games following a five-game winning streak.
"We scored early in the third period, and you can't sit back," Pavelski said. "They were pressing a little but we expected that, and there's no reason we shouldn't handle it."
Rinne gave Nashville a big lift early, stopping all 19 shots he faced in the first period.
"He's a great goalie. He's tough to beat," Couture said. "He plays well every time."
Rinne made a tremendous save during a Sharks power play during the final minute of regulation.
"They made a nice play to Pavelski," Rinne said. "Luckily, I was able to make that save. It was obviously pretty big at the time; just 20 or 30 seconds left in the third."
Legwand gave the Predators a 1-0 lead just over a minute into the second period when he slapped in a rebound of a shot by Hornqvist. The puck glanced off Sharks defender Torey Mitchell and into the net.
"The first one was just a puck thrown at the net and a lucky bounce off their pads," Legwand said. "We just kept battling, and to come up with the penalty kill in overtime and the final minute of the third period."
The Sharks scored the next three goals. Pavelski started the spurt by turning a nice backhanded pass from Couture into a goal just under five minutes into the period.
Havlat's power-play goal was the result of persistence. Couture and Ryane Clowe both had shots that bounced off Rinne's pads, and Havlat poked at the puck twice before wristing a shot into the top left corner.
Couture gave the Sharks a 3-1 advantage in the first minute of the third period.
Hornqvist got the Predators within 3-2 when he knocked in a loose puck from the crease. Suter tied it with a power-play goal a few minutes later.
NOTES: San Jose C Joe Thornton had two assists, giving him 703 and tying him with Jeremy Roenick for 49th place on the career list. ... Legwand has points in four straight games. ... Pavelski has scored a goal in six of his last eight games. ... Hornqvist has a career-best, five-game scoring streak. ... The Sharks outshot Nashville 19-7 in the scoreless first period.