Blake Geoffrion also tallied while David Legwand added an empty-net marker for the Predators, who have won four of five. Nashville also beat Detroit for the fourth time in 2010-11, clinching the victory in the six-game season series.
Pekka Rinne made 39 saves, including 20 in the third period.
Cleary scored on a tip-in in the final minute of the first period, during a power play.
But Nashville got the equalizer 8:42 into the second, on the rookie Geoffrion's third goal of the season.
Blum put the Predators ahead thanks to a heads-up play. Patric Hornqvist unloaded a shot on net from the right edge of the slot, and Howard made the save. But he couldn't control the rebound, which bounced out to the left side of the slot. Blum was there to one-time it across the goal line at 8:12.
The Red Wings had chances to score in the remaining time, but Rinne held strong, even after Detroit pulled Howard with a little more than a minute left.
Rinne stopped Tomas Holmstrom's rocket from the right boards with 37.1 seconds remaining, and shortly after the faceoff, Legwand scored to secure the win.
Nashville finished a five-game homestand, and plays at Buffalo on Sunday before starting a run of four games back at Bridgestone Arena...Detroit starts a five-game homestand Monday against Pittsburgh...Hornqvist had two assists.