Patric Hornqvist added an insurance goal with 2:21 left and David Legwand completed the scoring into an empty net as the Predators won for just the second time in seven games (2-4-1) to move back into a playoff position at the bottom of the tight Western Conference.
Roberto Luongo made 26 saves for the Canucks, who have alternated between a win and a loss for 12th straight games, but remain atop the overall NHL standings.
It's been a big week for Geoffrion, who scored his first NHL goal in his third career game Tuesday in Edmonton after becoming the first Tennessee native and the first fourth-generation player to make the NHL. His dad, Danny, played 111 NHL games, and both his grandfather, Bernie "Boom Boom" Geoffrion, and great grandfather, Howie Morenz, are in the Hockey Hall of Fame.
He scored the decisive goal Thursday by parking atop the crease unchecked and quickly snapping Jordin Tootoo's pass from the corner behind the net in before Luongo could get across his crease.
Hornqvist stole the puck from defenseman Dan Hamhuis breaking out of his own end, walked in alone and deked out Luongo for his team-leading 18th goal.
Rinne, starting for the 19th time in the past 20 games, won a battle between two of the NHL's top statistical stoppers by posting his fifth shutout of the season, though both goalies went through prolonged stretches of inactivity in a tight-checking game.
Rinne made a desperation save to take an empty net away from Daniel Sedin 11 minutes in, rolling over on his back after being faked to the ice and throwing out his glove arm. Luongo answered with a trio of saves, including two with the glove, during a rare flurry in front of his net 4 minutes later.
After the teams combined for just 10 shots in the first 15 minutes of the second period, Luongo denied Legwand from point-blank range before both teams took turns hitting the post.
First it was Mason Raymond ringing a backhand off the post past Rinne's glove with 4:09 left, then Tootoo struck iron over Luongo's glove on a 2-on-1 break exactly a minute later.
Rinne got another break 2 minutes after Geoffrion scored, as Henrik Sedin put a power-play rebound over his sprawled arm, through the crease and off the far post.
Notes: There was only one referee for the second half of the game after Justin St. Pierre sprained his ankle the second period. ... Canucks RW Daniel Sedin was held without a point for a third straight game for the first time this season, but he still leads the NHL with 79 points, one more than Tampa Bay's Steven Stamkos. ... Geoffrion's dad, Danny, scored his last NHL goal in Vancouver almost 30 years ago, on April 1, 1981. Nashville continues its four-game road trip with stops in Calgary and San Jose.