Only seconds into the overtime, Ward skated into the Detroit zone, attracting a pair of defenders. Ward passed and found Suter all alone. He skated in on Red Wings netminder Jimmy Howard and beat him with a shot from the right circle for the game-winner.
With the goal, Nashville made up for allowing the Red Wings to tie the game late in regulation when Pavel Datsyuk scored with 36.4 seconds left to force the extra frame.
Shea Weber, Martin Erat and J.P. Dumont also tallied and Pekka Rinne made 28 saves for the Preds, who moved two points ahead of the Red Wings for fifth in the West with two games to play.
Howard made 20 save as the Wings saw their seven-game winning streak snapped.