Rozsival, shooting first in the tiebreaker's fourth round, put a backhander past the Nashville goalie and Chicago counterpart Corey Crawford then wrapped up the win with a stop on Taylor Beck.
Patrick Kane and Brandon Saad scored in regulation for the Western Conference-leading Blackhawks, who notched their third win in four games.
Crawford finished with 25 saves for Chicago.
Beck found the net and David Legwand also scored for the Predators, who remained two points behind St. Louis for the eighth and final playoff spot in the West.
Rinne made 38 saves in the loss.