Gabriel Bourque and James Neal scored empty-net goals in the closing minutes for the Predators, who avoided their first two-game losing streak of the season.
Forsberg scored the 12th goal of his rookie season. Bourque sent a pass through the sticks of two Avalanche players, and the 20-year-old Swede broke into the open ice and flipped the puck over Calvin Pickard's left shoulder.
Rinne did the rest, stopping 26 shots. He has won eight of his past 10 starts, limiting opponents to two goals or fewer in those games.
Pickard made 33 saves in place of Semyon Varlamov, who missed his 10th game of the season after aggravating a groin injury during practice on Monday.
Stay Connected
NHL.TV™
NHL GameCenter LIVE™ is now NHL.TV™.
Watch out-of-market games and replays with an all new redesigned media player, mobile and connected device apps.
Introducing the new official NHL App, available for iPhone, iPad and Android smartphones and tablets. A host of new features and improved functionality are available across all platforms, including a redesigned league-wide scoreboard, expanded news coverage, searchable video highlights, individual team experiences* and more. The new NHL App on your tablet also introduces new offerings such as 60fps video, Multitasking** and Picture-in-Picture.
*Available only for smartphones
** Available only for suported iPads