Tootoo assisted on Nick Spalling's goal 12 minutes into the first period, which erased a 1-0 Anaheim lead, and Roman Josi's first NHL goal on a power play 2 1/2 minutes into the middle frame put Nashville ahead to stay.
Pekka Rinne turned aside 15-of-17 shots for the Predators, who broke a four-game home losing streak.
Andrew Gordon and Francois Beauchemin found the net for the Ducks.
Ellis finished with 25 saves.
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