Sven Baertschi, Derek Dorsett and Daniel Sedin, into an empty net, also scored for Vancouver (27-28-12), while Ryan Miller made 35 saves. Alexandre Burrows and Radim Vrbata chipped in with two assists each.
Roman Josi added two assists for the Predators, who came in on a 9-0-5 run and had not lost in regulation since Feb. 9, the longest active run in the NHL.
Despite the loss, Nashville still occupies the first wild-card spot in the Western Conference, seven points up on Minnesota and Colorado. Vancouver, meanwhile, is eight points back of the Wild and Avalanche.
Trailing 2-0 after a first period that saw the Canucks carry the play, the Predators came out with renewed vigour in the second and got on the board at 9:17 following a Vancouver turnover when Josi fed Fisher in the slot for his 11th of the season.
But the Canucks got that one back just 1:18 later off the rush when Yannick Weber fired a shot from the right side that bounced in off Horvat for his 11th after Rinne made the initial save.
The Predators responded with 3:40 left in the period when Josi wheeled through the neutral zone, played a give-and-go with Austin Watson, and found Sissons alone in front for his third.
Nashville outshot Vancouver 17-8 in the second and kept coming in the third.
Miller stretched to make a great toe save on a James Neal rebound effort to keep his team ahead five minutes into the third with the teams playing 4 on 4.
But that would be as close as the visitors would come before Sedin scored his 26th into an empty net with under a minute to go in regulation.
The Canucks dominated the first, outshooting the Predators 13-4, and were rewarded with two late goals to take a 2-0 lead to the locker-rooms.
Baertschi opened the scoring with 2:16 left when he fought off Shea Weber in front to swipe a rebound off the end boards past Rinne for his 14th.
Dorsett then doubled Vancouver's advantage with 58.3 remaining after taking a nice no-look feed from Bo Horvat before burying his fourth of the campaign and first in 30 games.
The Canucks promoted rookie winger Jake Virtanen to the top line with Henrik and Daniel Sedin for the first time since pre-season, but the 19-year-old was involved in an early fight and spent seven minutes of the opening period in the penalty box.
Notes: Dorsett's last goal before Saturday was on Dec. 20 against Florida. ... Henrik Sedin returned to the lineup after missing two games with an upper-body injury. ... Vancouver continues a four-game homestand on Monday against Winnipeg. Nashville wraps up a five-game road trip the same night in Edmonton.