The deal will pay Forsberg exactly $6 million in salary per season through 2021-22.
"To be honest with you, I never really thought about anything else but staying in Nashville," Forsberg said Monday during a conference call. "I think we've got something really interesting going and I wanted to be a part of that."
The 21-year-old Swede was a restricted free agent, and teams are always wary of offer sheets from competing clubs come July 1.
"Filip is among the most skilled, dynamic talents we've ever had in the organization and is an integral part of our success for the next six years and beyond," Predators general manager David Poile said in a statement. "While we have already seen his creativity, hockey sense and puck skills, he has yet to reach the peak of his abilities. We have full confidence that he will continue to blossom into one of the top players in the world.
"This is just another example of our ownership group's commitment to the franchise, our fans and the entire community to build a team that can reach our ultimate goal of winning the Stanley Cup."
Forsberg led the team with 33 goals and 64 points overall this past season.
Nashville advanced to the second round of the playoffs this past season before losing to the Sharks in seven games.
"It's going to be really interesting to be a part of this and just try to build on it," Forsberg said. "We had a really good team last year. If all the players that we've still got can just improve and get better, and we add something in free agency and stuff like that, we have a really good chance of winning the Cup in the future."
Information from The Associated Press was used in this report.