Henrik Zetterberg won a face-off in the Nashville zone and got the puck to Brian Rafalski. The Red Wings defenseman took a slap shot from the left point that Cleary redirected past Pekka Rinne, clinching a post-season berth for the Wings for the 20th consecutive year.
Detroit trailed 3-0 early in the second period but rebounded on goals from Jakub Kindi and a pair from Justin Abdelkader to force an extra frame.
Jimmy Howard turned aside 31-of-34 shots for the Red Wings, who reached the 100-point plateau for the 11th straight season.
Ryan Suter, Martin Erat and Sergei Kostitsyn found the net for the Predators, who are seeded fifth in the Western Conference playoff standings with three games remaining.
Rinne absorbed the loss despite making 37 saves.