Ryan Johansen and Calle Jarnkrok also scored in the second period as Nashville built a 4-goal lead.
Artturi Lehkonen and Lukas Vejdemo had goals in the third period for Montreal, which lost its third straight game.
Juuse Saros made 32 saves for the win.
Going into the game, Saros hadn't allowed a goal since losing to Minnesota on March 3. Lehkonen's goal brought an end to Saros's streak, which included consecutive shutouts against Dallas.
''It's been unbelievable,'' Johansen said. ''At one point, it felt like the kid's going to keep going and going. He's really stepped up for us.''
Carey Price stopped 32 shots for the Canadiens in his league-leading 58th game of the season.
A third-period rally wasn't enough to make up for a four-goal lapse in the second period for Montreal.
''Just as professionals, you've got to be better than that,'' forward Brendan Gallagher said. ''It was extremely disappointing, obviously. We felt like we left our goaltender out to dry. You have a second period like that, it costs you the game.''
Canadiens coach Claude Julien called the team's play in the second ''catastrophic.''
The Bell Centre hosted a pregame ceremony memorializing Habs great Henri Richard, who passed away on Friday. The crowd gave a standing ovation to Richard's family, which was in attendance, followed by a moment of silence.
Johansen opened the scoring less than a minute into the second, rifling a wrist shot past Price for his 14th of the season.
Just over two minutes later, it was Jarnkrok's turn. With Price screened off the faceoff, he whipped a quick shot into the net.
With Joel Armia in the penalty box on a tripping call, Forsberg took a pass, skated just outside the face-off circle and put the puck over Price's outstretched glove for his 20th of the year. That gave Nashville a 3-0 lead 8:18 into the second.
Forsberg struck again at the 16-minute mark, though this time luck played a role. A Roman Josi point shot hit a skate in front, bouncing directly onto the Swede's stick, leaving Price out of position to make a save.
The Canadiens had little response to the offensive onslaught. One of their few chances came when winger Lehkonen broke in on a 2-on-1 with rookie Nick Suzuki, but shot the puck harmlessly into Saros' glove.
Nothing went right for the Canadiens in the period. Even the Bell Centre's lights got in on the action, with some above the ice flickering off and forcing a stoppage during a rare moment of action in the Preds' zone.
Lehkonen finally managed to get his team on the board early in the third, shoveling a rebound off the boards past Saros.
Vejdemo posted his first NHL point in his seventh game. The rookie worked his way to the front of the net and one-timed a Dale Weise pass from behind the goal line in.
The Canadiens kept the pressure on, but Saros came up big on several in-close chances with Price pulled for the extra attacker.
Price kept his team in the game in the first, stopping all 13 shots he faced.
Early penalties to Ben Chiarot and Xavier Ouellet kept Saros's end of the ice quiet - other than two shots with Ouellet off the ice, Montreal wouldn't get another puck on net until the 12-minute mark. In total, Saros was challenged just seven times in the first, though a late Canadiens' surge resulted in several spectacular saves.
It was Montreal's third loss in a row as the Canadiens were coming off two disappointing games in Florida, getting shut out 4-0 by the Tampa Bay Lightning on Thursday and losing 4-1 to the Panthers on Saturday.
The Predators are locked into a tight battle for the West's two wild-card spots. They entered the game tied with the Vancouver Canucks with 76 points. Minnesota and Winnipeg hold the wild cards with 77 and 78 points, respectively.
NOTES: Habs forwards Jonathan Drouin and Tomas Tatar and defenseman VIctor Mete sat out with injuries while defenseman Karl Alzner was a healthy scratch. ... Nashville defenseman Dan Hamhuis sat out with an injury while Austin Watson, Yannick Weber and Colin Blackwell were healthy scratches. ... Montreal lost its third in a row as the Canadiens were coming off two disappointing games in Florida, getting shut out 4-0 by the Tampa Bay Lightning on Thursday and losing 4-1 to the Panthers on Saturday. ... The Predators are locked into a tight battle for the West's two wild-card spots. They entered the game tied with the Vancouver Canucks with 76 points. Minnesota and Winnipeg hold the wild cards with 77 and 78 points, respectively.
Predators: At Toronto Maple Leafs on Thursday.
Canadiens: Host Buffalo Sabres on Thursday.
For more AP NHL coverage: https://apnews.com/NHL and https://twitter.com/AP-Sports