Calle Jarnkrok, Mike Ribeiro, and James Neal also scored for Nashville, who stopped a two-game losing streak. Shea Weber had three assists.
Oliver Ekman-Larsson and Kyle Chipchura had the goals for Arizona, who saw their three-game winning streak end.
Nashville lost center Mike Fisher at 7:56 of the second period to a lower-body injury. In the Arizona zone, Martin Hanzal checked Fisher into the left boards. Fisher left the ice without putting any weight on his right leg.
Nashville took a 3-2 lead in the third on Forsberg's wrist shot.
He scored his second at 17:01 and Neal added an empty-net goal late.