Kimmo Timonen skated around the back of the net and backhanded a pass to Sullivan in the slot, and he beat Ilya Bryzgalov for his 13th goal this season. The victory was Nashville's first against Anaheim since Oct. 8, 2005 when the Predators won in a shootout.
This game featured two of the top three teams in the NHL. But the Ducks, who came in with a league-high 64 points, lost for the fifth time in six games.
Timonen, Paul Kariya, David Legwand and Jean-Pierre Dumont also scored for Nashville. Marek Zidlicky had three assists and Martin Erat added two assists.
Teemu Selanne and Ryan Getzlaf each scored for Anaheim, Dustin Penner added a goal and an assist, and Corey Perry had two assists.
The game marked the return of Nashville goaltender Tomas Vokoun, who had missed 21 games since tearing a ligament in his left hand Nov. 23.
He gave up a goal on the first shot by Chris Kunitz. But he stopped Selanne from up close in overtime and turned aside 27 of the 31 shots he faced.