"The whole mentality is changing, our whole attitude," Giguere said. "We seem to believe in ourselves a little bit more. It is a different voice, and the guys seem to be responding to it right now."
The Avalanche are 4-0 under Roy, their best mark since the franchise relocated from Quebec to Colorado for the 1995-96 season.
"We are trying to make the playoffs and go as far as we can," Avalanche forward Ryan O'Reilly said. "We are approaching every game to learn and get better, and right now we are having success and doing it."
Giguere, in his first start of the season, made several sparkling saves. His best came midway through the second period when he stoned Patrice Bergeron on a 2-on-1 rush.
O'Reilly gave the Avalanche the lead in the final minute of the first period and Matt Duchene sealed the win with an empty-net goal in the closing seconds.
Andre Benoit and Duchene assisted on O'Reilly's goal.
Tuukka Rask stopped 28 shots for Boston and has allowed only three goals in three games. Boston was blanked for the first time in 72 games.
"Sometimes you run into a hot goaltender, a team that's playing well ... and they did tonight," Bruins coach Claude Julien said.
The Colorado goalie tandem of Giguere and Semyon Varlamov has allowed three goals in the club's four wins under Roy, a Hall of Fame goalie during his playing career.
O'Reilly put Colorado ahead 1-0 with a power-play goal with 41 seconds left in the first. He threw the puck toward the net and scored through traffic with a shot that Rask never saw.
Duchene scored into an empty net with 25.7 seconds remaining after he hit two posts on another rush with under three minutes remaining.
Colorado extended its point streak in Boston to eight games (6-0-2). It is the longest current winless streak against any opponent for Boston.
"Their goalie obviously did a great job," Bruins defenseman Torey Krug said. "Shame on us for not scoring. We've got to step it up a bit."
The Bruins will play at Columbus on Saturday in their first road game of the season. Colorado will wrap up its three-game road trip at Washington.
Boston allowed a power-play goal for the first time in eight short-handed situations this season. ... Colorado has killed all 12 power plays against. ... Bruins LW Milan Lucic received a 10-minute misconduct in the second period for roughing Gabriel Landeskog and knocking him down twice. ... Boston has never won an overtime game at home against the Quebec/Colorado franchise. ... Cody McLeod played in his 400th game for Colorado. ... Jarome Iginla and Loui Eriksson have yet to record a point for the Bruins.