Forsberg’s second goal of the game tied it 3-3 at 19:23, swatting the puck out of the air near the left goal post, and Trenin scored on a rebound with 22 seconds left for the 4-3 final.
Jeremy Lauzon scored, and Juuse Saros made 25 saves for the Predators (7-10-0), who have won their past two games after losing four in a row.
Nashville outshot Colorado 15-6 in the third.
Valeri Nichushkin had a goal and an assist, and Cale Makar had three assists for the Avalanche (11-6-0), who had won three straight. Alexandar Georgiev made 27 saves.
Andrew Cogliano scored short-handed to give Colorado a 1-0 lead at 14:00 of the first period. Nichushkin took the initial shot on the rush, and Cogliano scored on the rebound.
Forsberg tied it 1-1 at 19:42 on a wrist shot from the left face-off circle after a backhand feed from Cody Glass made a backhand pass.
Lauzon gave Nashville a 2-1 lead at 3:56 of the second period on a wrist shot from the point.
Devon Toews scored for the Avalanche to tie it 2-2 at 8:55 of the second period on a one-timer from the point on a pass from Nathan MacKinnon.
Nichushkin gave Colorado a 3-2 lead on a power play at 13:33 on a redirection of Makar’s shot.
The Avalanche outshot the Predators 14-6 in the second.
