The Predators got off to a fast start as Wilson, skating in the middle of the slot, scored on a tip-in of a pass from Sergei Kostitsyn at 12:22 of the opening period.
Kostitsyn has one assist in each of Nashville's four games.
Smthson, stationed a few feet from the net on the left side, put a wrist shot behind New Jersey goaltender Johan Hedberg at 13:53.
Hedberg stopped a penalty shot by Kostitsyn at 11:53 of the first period.
Neither team scored in the second period, but one minute into the third and on a power play, Kovalchuk fired a wrist shot high into the left corner of the net that bounced off of the crossbar.
"It's always nice to score like that," Kovalchuk said. "It gives you a lot of energy."
Kovalchuk played a game-high 32:04. Nashville coach Barry Trotz stressed that it is difficult for a player to be on the ice for that length of time.
"He can fly and he is a big man," Trotz said. "He's hard to contain. Their whole team got momentum after his first goal."
Clarkson tied it 2-all at the 7:57 mark, beating Rinne from his glove side.
"Guys were screaming between periods that this was our game," Clarkson said. "We have enough guys in this room that believe. It's fun to come back like this."
New Jersey coach Peter DeBoer thought his team outplayed the Predators despite the rough start. New Jersey out shot Nashville 41-31, the fourth time this season the Predators have been outshot.
"I think we deserved to win," DeBoer said. "Their goaltender made a few big saves early to give them a chance to have the lead they had. We felt really good about our game and we just needed to stick with it. That is what we did."
Trotz saw some reasons to be happy with his team despite the second home loss in as many outings.
"We let it get away by a hair," Trotz said. "There are so many positives out of the whole deal. We saw a little bit more of Predator hockey, especially in the first period."
Material from wire services and team media was used in this report.
1 - 0 NSH
2 - 0 NSH
2 - 1 NSH