Forsberg, who also had an assist, stripped the puck from Andre Burakovsky behind the net, skated to the front and lifted the puck past Chicago goalie Spencer Knight for his second goal of the game.
Juuse Saros, who missed the previous two games with an upper-body injury, returned to the net and made 27 saves for the Predators (33-28-9), who have won four in a row and five of seven (5-1-1).
Nashville moved three points ahead of the Los Angeles Kings for the second wild card into the Stanley Cup Playoffs from the Western Conference.
Connor Bedard scored, and Spencer Knight made 30 saves for the Blackhawks (26-32-13), who have lost four of five (1-2-2).
Nick Lardis gave the Blackhawks a 1-0 lead at 4:34 of the second period. His wrist shot from the left face-off circle deflected into the net off the stick of Predators defenseman Nicolas Hague.
Forsberg tied it 1-1 at 6:31, scoring from the slot after taking a pass on a give-and-go from Jonathan Marchessault, who fed it from the right circle.
Bedard gave Chicago a 2-1 lead at 13:57 when he took a stretch pass from Ryan Greene, moved in on Saros and scored after going backhand to forehand.
Steven Stamkos tied it 2-2 at 9:43 of the third period, banking the puck in off the right post after getting a feed from Forsberg in the left circle.
The Blackhawks played most of the game without defenseman Matt Grzelcyk, who left the game at 4:49 of the first period with an apparent upper-body injury.