Klein's tally 5:15 into the middle frame put the Preds ahead for good at 2-1 and Colin Wilson and Sergei Kostitsyn followed with markers to ice the victory, Nashville's 10th in 11 games.
Chicago saw its three-game winning streak snapped despite two goals from Marian Hossa.
Corey Crawford allowed four goals on 17 shots before being pulled in the second period.
Ray Emery relieved and made 5-of-5 saves the rest of the way.
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