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. It was Spezza's first point of the season.
Spezza has at least one point in his last five games against Nashville, with six goals and two assists in that span.
The Stars outshot Nashville 16-4 in a scoreless first period, and then Cracknell made it 1-0 2:55 into the second.
Johnny Oduya's shot from the left point was first deflected by Curtis McKenzie in the high slot and then tipped again by Cracknell, who was standing just to the right of Rinne.
Rinne finished with 35 saves.
Fisher tied it 4:27 into the second on the power play.
From the left corner, Filip Forsberg found Fisher in the low slot, where he snapped a wrist shot past Lehtonen.
Five of Nashville's seven goals this season have come via the power play.
Fisher succeeded Shea Weber as Nashville's captain after Weber was traded to Montreal in the offseason in exchange for P.K. Subban.
NOTES: Nashville's opponents have scored first in each of their three games this season. ... The Predators played without D Anthony Bitetto and RW Miikka Salomaki, who are both sidelined multiple weeks with upper body injuries. ... Spezza has 11 goals and ten assists in 19 career games played against Nashville.