Wilson scored the game-winning goal with 3:27 remaining. He took the puck from defenseman Victor Bartley on a tic-tac-toe play and scored his 18th goal of the season.
"I think over the last two games, I've felt good and have been getting my chances," Wilson said. "They haven't been going in, so it was just nice to get that in the third period."
Wilson had a career-high seven shots on goal, including five in the first period. He said he started the game with an extra emphasis on shooting the puck on net.
"I think I did more in the first," Wilson said. "I kind of got away from that. [Nashville coach Peter Laviolette] was on me making sure I started putting pucks to the net again. Just trying to get there."
Smith gave the Predators a 3-1 lead with 2:21 remaining in the third period on a wraparound. He worked to retrieve the puck behind the net and was able to put it past Devils goaltender Cory Schneider for his 15th of the season.
Laviolette shuffled all four lines to add more balance, even though the Predators had won four consecutive games heading into Saturday. The changes resulted in 38 shots on goal and a fifth consecutive win.
"Sometimes when you're cruising along, you just want to stay the course and not rock the boat too much," Laviolette said. "Something told me it just wasn't right, and the guys knew it too. As I approached the players about it and we talked about it, they felt the same way too that we weren't on the attack. It was just different. It's good to see different combinations. It was a good balance tonight is what it was."
Predators goaltender Pekka Rinne got his NHL-leading 33rd win and improved to 14-1-1 in his past 16 games.
"[Rinne] has been great," Smith said. "He's great every night. He's going to make the big saves. He's a big guy back there making plays."
Nashville (38-12-6) improved to 24-3-1 at home and have won 13 of the past 14 games at Bridgestone Arena.
New Jersey (21-26-9) has lost four games in a row and has scored five goals in that span.
"We withstood the storm at the beginning of the [first] period," Devils co-coach Lou Lamoriello said. "They came at us, we knew they would, and Cory gave us a chance. The effort and the execution in the last two periods, you can't ask for any more. We had opportunities. We did a great job killing penalties in the middle of the first period, we got a power-play goal and then 5-on-5 we played it even the whole night with more chances than they did at the end. We missed an assignment and it was 2-1."
Matt Cullen gave Nashville a 1-0 lead at 8:49 of the first period on a wrist shot from point-blank range. Cullen took a pass from Smith behind the New Jersey net and put it past Schneider. The goal was Cullen's fifth of the season and his first in 19 games.
"Anybody can play with anybody," Smith said. "I think everybody knows their role. It's up to the coaching staff to kind of move us around to whatever slot they think we should be put in, and then go play. I think that's kind of the theme of this team. We go out and play and play for each other, and good things have been happening."
New Jersey forward Steve Bernier tied the game 1-1 at 17:37 of the second period on the power play. Bernier was able to swat a rebound past Rinne for his 10th of the season.
Predators defenseman Mattias Ekholm returned to the lineup after missing two games with an upper-body injury. He was a plus-2 in 17:15 of ice time.
"Excellent," Laviolette said. "He didn’t miss a beat, and we missed him. It was good to get him back."
Nashville outshot New Jersey 21-3 in the first period. Schneider made several saves from point-blank range to keep the Devils in the game.
Schneider made 33 saves on Friday against the Chicago Blackhawks and made 35 saves on Saturday.
"It’s fun to play down in the offensive zone," Smith said. "That shows you're doing something right, and we kind of harped on that going into the first period and coming out and having a good one. We did, and it was great. We were moving our feet, creating opportunities, shooting. Our puck retrievals were great."