Spaling, who also scored in the first period, was waiting at the doorstep to put in a loose puck after Ducks goalie Ray Emery poke-checked it away from an onrushing Jordin Tootoo, giving the Predators a 3-2 lead 4:53 into the final frame.
Nashville goalie Pekka Rinne -- who recorded 25 saves -- made the advantage hold up later by blanking Anaheim on a power play and the Predators added an empty-netter from David Legwand to ice the victory.
Steve Sullivan also scored for Nashville, which won its first-ever, post-season series after going winless in five previous playoff appearances and now advances to the Western Conference semifinals.
Jason Blake and Teemu Selanne found the net and Emery finished with 26 saves for Anaheim.
Swim Week Miami Beach 2014 [PHOTOS]