Price had Burns' initials on his mask as he made 30 saves in a 2-0 victory over the rival Toronto Maple Leafs on Saturday night.
''He was a special person and he did a lot of great things in this league for both teams,'' Price said of the 58-year-old who had success as coach in Montreal, Toronto, Boston and New Jersey before his illness drove him to step down in 2004.
''He left me a message before the season started and I was really touched. He gave it to (assistant coach) Kirk Muller and he passed it on to me.''
Asked what Burns said, Price just said: ''That will always be here with me.''
Perhaps it has contributed to what is becoming a breakout season for the Canadiens' fifth overall draft pick from 2005.
Price's fourth shutout matched his total from his first three NHL seasons combined. His 12th win is one short of his total for all of last season, when he went 13-20-5 and lost the starting job to Jaroslav Halak, who was traded in the summer to the St. Louis Blues.
Jeff Halpern scored short-handed and Michael Cammalleri got one on a power play for Montreal (13-6-1), which was rebounding from a 3-0 loss to Nashville that followed a 3-0 win over Philadelphia. It was Price's third shutout in his last six starts. The Canadiens have won five of their last six games.
Jonas Gustavsson made 37 saves for the Leafs (7-9-3), who ended a two-game winning run. Several of Gustavsson's saves were spectacular as the Canadiens enjoyed numerous odd-man rushes.
Toronto was shut out for the fourth time this season.
The Canadiens also cooled off Toronto's power play, which had scored six times in the last two games but went 0-for-3 for the night. Price and his defence, particularly the pair of Roman Hamrlik and Jaroslav Spacek, were instrumental in frustrating the Leafs while playing a man down.
''Last year saw tremendous growth for Carey Price,'' said coach Jacques Martin. ''He had a tough tie believing that what was happening to him was going to be good for his career.
''Maturity and composure have been key to his success.''
The Canadiens avenged a 3-2 loss in Toronto in the opening game of the season for both clubs on Oct. 7.
The night began with a scoreboard tribute and moment of silence for Burns, which for a brief moment had the competing groups of Montreal and Toronto fans in the Bell Centre crowd of 21,273 cheering in unison.
The Canadiens had not scored since 6:24 of the third period against Philadelphia on Tuesday night before Halpern broke the drought at 9:17 of the second frame with his fifth of the season.
The veteran centre won a faceoff in the Leafs' zone from Mikail Grabovski and then deflected Jaroslav Spacek's point shot in on their 26th shot on Gustavsson.
Former Canadien Mike Komisarek, who was booed each time he touched the puck by Montreal fans, was the goat on the Canadiens' second goal. He fired a backhand pass across the front of his own net while killing a penalty and Cammalleri slammed it in from close range. The goal came on Montreal's only power play.
''It was a tough play,'' said Komisarek. ''You try to hit the guy in the middle and it passed between two guys.
''It was the right play. I just have to execute it better.''
Komisarek also had a thought for Burns.
''It was emotional for both teams,'' he said. ''Pat Burns meant a lot to our organization and the Canadiens.
''The hockey world lost a great hockey mind. The tribute was very tastefully done.''
Notes: It was Brian Burke's 1,000th career game as a general manager in Hartford, Vancouver, Anaheim and Toronto. . . Defenceman Yannick Weber, called up from Hamilton this week, played on left wing in place of Mathieu Darche for Montreal. Dustin Boyd was scratched. . . Brett Lebda and Luca Caputi were out of the Leafs lineup. James Reimer backed up Gustavsson in place of Jean-Sebastien Giguere (groin).
1 - 0 MTL
2 - 0 MTL