The deal will pay Forsberg exactly $6 million in salary per season through 2021-22.
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 GM 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 overall 64 points this past season.