Maxim Lapierre scored in the second period to put the Canadiens up 3-0, Michael Ryder scored with 2:21 remaining in the third and Montreal ended New Jersey's three-game winning streak.
The final minute of the game was played throughout a sustained ovation which was punctuated by a load roar from the customary crowd of 21,273 at the final siren.
"It feels good," Price said. "I wish they could have cheered for everybody like that. Everybody deserved an ovation like that tonight."
The 20-year-old rookie improved to 5-2 in seven starts since taking over the No. 1 job when Cristobal Huet was sent to Washington at the trade deadline.
"It's nice to get their support," Price said. "It definitely gets you motivated when they're doing that. It's pretty tough to not be ready for the next shot when you're hearing those kind of screams."
Price turned aside 20 shots in the first period and 13 in the second as the Canadiens built a lead despite being outshot 33-18 in the first two periods.
"He was great," Devils forward Zach Parise said. "He made some unbelievable saves at big parts of the game, when it was 1-0. He was awesome, what else is there to say? He definitely came to play tonight."
Montreal, which finished a 2-2 Western road trip with a 3-1 loss in Anaheim on Sunday, leads the Northeast Division with 87 points.
"That was really important, shutouts aside," Price said. "We definitely needed the two points and we're happy to be in first place but we know it's going to be up and down for the rest of the year and we're going to be fighting really hard for first place every time we step on the ice.
Rookie Sergei Kostitsyn set up the Canadiens last two goals as Montreal won its third in a row over the Devils to finish 3-1 and win its first season series against New Jersey in 15 years.
Martin Brodeur made 29 saves in his 28th start in a row for the Devils, who remain atop the Atlantic Division with 86 points, one ahead of idle Pittsburgh.
Smolinski opened the scoring 4:51 in with his sixth goal of the season to end an 18-game goal drought.
The veteran center scored his first goal since he beat Brodeur twice in a 4-3 win in New Jersey on Jan. 24. He received Tom Kostopoulos' back pass off the right boards before beating the Devils goalie with a wrist shot inside the right post.
"It was just a bad read for me," Brodeur said. "He was coming down the wing and he was looking the whole way to make a pass and he changed his mind and caught me leaning more than cheating and he caught the sweet spot. You've still got to make that shot but it's one that I'd like to have back, for sure."
Koivu made it 2-0 at 18:28 when he beat Brodeur on a give-and-go passing play with Kostitsyn. The Canadiens captain passed right to the 20-year-old rookie before putting Kostitsyn's return pass into an open left side for his third goal in four games.
"We were going in 3-on-2 and I just saw that Saku was open," Kostitsyn said.
Kostitsyn made another fine play to increase Montreal's lead to three on Lapierre's seventh goal 14:20 into the second.
Lapierre had a wide-open right side to shoot at when he received Kostitsyn's centering pass and the Canadiens right wing easily beat Brodeur's lunging save attempt with a low shot under the goalie's glove.
