Adam Cracknell added his second goal of the season for Dallas, and Kari Lehtonen made 27 saves.
Mike Fisher scored for Nashville early in the second period to tie it 1-1. The Predators have lost two straight.
Spezza's tiebreaking goal came 9:15 into the third. Antoine Roussel had the puck in the right corner and sent it out front, where Spezza was able get his stick on it and tip it past goaltender Pekka Rinne.
Rinne made 35 saves.
