Roman Josi lobbed a pass from the Nashville defensive zone, and Duchene received it at the opposite blue line, cut to the net and chipped the puck over goalie Alex Nedeljkovic's glove.
"Enjoy it tonight, take the good vibes and move forward, and keep going at the task at hand here," Duchene said.
Carolina leads the best-of-7 series with Game 4 at Nashville on Sunday (2:30 p.m. ET; NBCSN, SN, TVAS, BSSO).
Ryan Ellis had a goal and two assists, and Juuse Saros made 52 saves for the Predators, who are the No. 4 seed in the Discover Central Division.
"It feels really good," Nashville coach John Hynes said. "It's nice we found a way to win and get ourselves back in the series. For the players, the fans and the organization, it's a good night."