Price outdueled counterpart Henrik Lundqvist, and earned a key win on the strength of Max Pacioretty's goal with 4:17 left that lifted Montreal over New York 1-0 on Thursday night.
"I actually thought about that in the first intermission," Price said. "Hank was playing really well. Both teams were playing really strong defensive hockey."
Pacioretty sent a wrist shot from above the right circle between the legs of Rangers defenceman Ryan McDonagh that appeared to surprise Lundqvist before squeezing inside the right post for his 22nd goal.
"I don't expect him to shoot," Lundqvist said. "When I am about to push to my left, I got stuck in the post. There is no excuse. I have to expect him to shoot, even though it's a tough angle.
"I have to stop it. This is on me, and it's a tough feeling."
That goal gave Price his 26th win and third shutout of the season in a stellar 24-save effort. Montreal has won four in a row — three with Price in net.
Lundqvist made 25 saves for the Rangers, who have lost two straight following the All-Star break and managed only one goal. They were beaten 4-1 by the Islanders on Tuesday.
"An inch there, an inch here, we might have been able to beat their goaltender," Rangers coach Alain Vigneault said. "They got the goal that made the difference."
The Canadiens, eliminated by the Rangers in six games in last year's Eastern Conference finals, lead New York by seven points in this season's playoff race.
This rematch had a post-season feel to it.
"I am just worried about doing my job — stopping the next shot," Price said. "It's fun to win games 6-5, but if we want to be successful, we're going to have to really bear down on those low-scoring checking games.
"Tonight we were the team that got the lucky bounce."
The netminders took turns in making spectacular saves. Lundqvist snared a shot by Pacioretty with a lunging reach of his glove in the first period that had him staring skyward.
"I've shot on him a lot," Pacioretty said. "He baited me with the glove and I felt pretty dumb, I took the bait. He was waiting for it. I feel like I could've shot anywhere else on the net and I would've scored."
Price shined in the third period, first when he denied Rick Nash on a short-handed breakaway, and then when he turned aside Martin St. Louis with a twisting, sprawling save that left him flat on his back.
"I just kind of reacted," Price said. "I got a shoulder and an ear on it. I thought it went behind me, so I spun around, and it ended up underneath me."
Montreal thought it took the lead with 1:56 left in the first when former Ranger Dale Weise ripped a shot that Lundqvist stopped. A video review ensued, and it appeared Lundqvist's pad was in the net at the time, but there was no conclusive evidence.
The opening period was particularly testy, a somewhat unusual scenario considering the teams hadn't met since New York's 5-0 home victory on Nov. 23. The Canadiens won 3-1 in the first game of the season series on Oct. 25.
While there was only one fight in the first period — a bout between former Rangers forward Brandon Prust and Tanner Glass with 1:22 remaining — there were several other skirmishes and a near-fight between New York's Chris Kreider and physical Canadiens defenceman P.K. Subban.
The two came together with 6:55 left. Kreider was called for unsportsmanlike conduct, and Subban was whistled for embellishment. They then jawed at each other as they left the box.
Kreider dropped his gloves and grabbed Subban, who kept his gloves on. The two returned to the penalty box because of more unsportsmanlike conduct.
"I knew it was going to be an intense game," Subban said. "You have two teams that are probably two of the best teams in the conference. We wanted to make sure we came out with our best performance."
The first period ended with a rough scrum along the boards in the Rangers end that featured Montreal's Lars Eller throwing punches at Derek Stepan. Both were sent off for roughing, but Eller received an extra 2 minutes that gave New York its first power play.
NOTES: Price has allowed two goals or fewer in 14 of his past 15 appearances. ... RW Lee Stempniak and D John Moore rejoined the Rangers lineup after being healthy scratches against the Islanders. They replaced C J.T. Miller and D Matt Hunwick.