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Price got his 13th win to match his total from last season.
"It's just a credit to our guys," said Price, who has started all but one of the Canadiens' 22 games. "Everybody's on a roll right now and everybody's doing what they have to do to get wins."
Lars Eller scored his first goal in 21 games with Montreal late in the second to put the Canadiens up by three. Mike Cammalleri and Tomas Plekanec also scored to help the Canadiens win for the sixth time in eight games.
Alec Martinez scored his first NHL goal in his season debut for Los Angeles, which has lost five of six after a 12-3 start.
"Right now there's a little bit of frustration setting in because this thing keeps rolling and we've got to work through it," Kings coach Terry Murray said. "That's all there is to it."
Jonathan Bernier made 29 saves in his seventh start of the season. The Montreal native fell to 0-5 on the road with several dozen members of his family and friends looking on.
"Obviously, I was a little bit nervous but I knew what was coming up and I felt pretty good," Bernier said.
Chosen fifth overall in the 2005 draft, Price has four shutouts in 21 starts this season.
The 23-year-old goalie went 13-20-5 and had no shutouts in 41 games in 2009-10. He lost his starting job to Jaroslav Halak, who was traded in the offseason despite leading the Canadiens to the Eastern Conference final. Halak was sent to St. Louis in the deal that brought Eller to Montreal.
"He's obviously on top of his game. I think he's their best player right now, in my mind, from watching highlights and stuff," said Kings left wing Ryan Smyth, who drew an assist after he got tangled up with Price on the play that led to Martinez's power-play goal 1:57 into the second.
Kostitsyn, who scored a power-play goal midway through the first, assisted on Plekanec's goal that made it 3-1 at 3:16.
Bernier stopped Kostitsyn's shot and Canadiens captain Brian Gionta picked up the rebound, circling patiently before feeding a pass to Plekanec, who one-timed a drive into the right side of the net to put the Canadiens up 3-1 moments after Martinez had drawn the Kings within one.
"He did a good job of finding that seam in the slot and it was pretty easy laying it to him," Gionta said.
Eller, who had one assist through his first 20 games, got his first Montreal goal with 1:57 left in the second when he put away a rebound of Alexandre Picard's shot off the right post.
"I'm sure I've got a few text messages waiting for me and some calls, too," said Eller, who noted that his parents watch almost all of his games at home in his native Denmark.
Martinez drew Los Angeles to 2-1 during with a power-play goal early in the second. Price complained in vain after the goal because he got tangled up with Smyth, who fell to the ice after Justin Williams' shot struck him, leaving Martinez with the rebound for a backhand shot into an open right side.
"I think he took a shot in the ankle or something and he kind of went down and my pad got hooked on his leg, and I didn't have a chance to get across," Price said. "I don't know why it wasn't called, but whatever."
Kostitsyn opened the scoring 9:59 in when he drove the right side on a 2-on-1 and fired a shot past Bernier for his eighth goal.
"My defenseman put his stick in the way and that bothered me a bit, but that was a really good shot through the five-hole," Bernier said.
Murray said, "I didn't like that goal." Asked to explain what he didn't like about it, he laid the blame squarely on Bernier.
"I don't like it going through goaltenders," he said.
Cammalleri made it 2-0 with a drive from the point for his seventh during a 4-on-4.
Bernier has yet to win a road game in his career. He is 6-0 at home, including four wins at the beginning of the 2007-08 season before he was returned to his junior team. ... Martinez was recalled from Manchester of the AHL on Monday. He had no points in four games last season. ... Price had a career-high 24 wins as a rookie in 2007-08. He won 23 games the following season.