Colorado remained unbeaten in four games, having given up just three goals for the season.
Giguere is starting his 15th NHL campaign and his third with Colorado. This was his 37th career shutout.
Boston lost for the first time in its three contests and suffered a blanking after producing seven goals in its first two games.
Ryan O'Reilly gave Colorado the lead with 40 seconds left in the first period on a power-play goal. Giguere protected the 1-0 advantage until Matt Duchene scored into an empty net with 26 seconds to go in the game.
Tuukka Rask allowed one goal on 29 shots. He has given up only one goal in each of his three starts.