Roman Hamrlik and Tomas Plekanec scored for the Canadiens, who have lost four straight since opening the season with a pair of overtime wins.
Plekanec drew Montreal even at 2-2 when he took Andrei Kostitsyn's centering pass and put a shot through Anderson's pads 8:26 into the third.
The Canadiens, who celebrated their 100th season last year, did not play any games in 2004-05, when the league's schedule was ultimately canceled because of a work stoppage.
Price, who faced the Avalanche for the first time, stopped 20 shots. He debuted a new mask featuring six "Ace of Spades" cards, yet he was decidedly unlucky in the second.
Cumiskey got credit for his first goal since January 2007 as Colorado banked in two shots off Montreal defensemen in the second period to take a 2-1 lead.
Hejduk drew the Avalanche even at 1-1 during 4-on-4 play 13:39 into the middle period. He got credit for his second goal of the season - 10 seconds after Plekanec was sent off for cross-checking - when his shot struck Josh Gorges' left skate, sending the puck skidding slowly between Price's pads and into the net.
Price had little chance on Colorado's second goal at 19:32. Cumiskey's shot from the left side struck Shawn Belle, who was called up from Hamilton of the AHL on Tuesday, and caromed past Price to give the Avalanche a 2-1 lead with 28 seconds left in the second.
Hamrlik scored his second goal of the season on a power play midway through the first. Canadiens forwards Kostitsyn and Plekanec exchanged the puck in the right corner and Plekanec moved up to feed a centering pass to Hamrlik in the slot, who put a shot along the ice into the open left side.
NOTES: Cumiskey, drafted 222nd overall by the Avalanche in 2005, had one goal in his previous 59 games. He scored his first and only other NHL goal on Jan. 6, 2007 against Minnesota's Manny Fernandez. ... Belle appeared in nine games with Minnesota in 2006-07. ... The Canadiens are 56-27-16-1 overall in home openers, including two wins and six losses during their first eight seasons in the National Hockey Association before they joined the NHL in 1917-18 as one of the league's founding franchises. ... Montreal went 24-10-7 last season at the Bell Centre.
1 - 0 MTL
1 - 1 Tie
2 - 1 COL
2 - 2 Tie
3 - 2 COL
Too many men/ice - bench