Anderson had 17 of his saves in a one-sided first period and tough-guy Cody McLeod and T.J. Galiardi added goals to help the Avalanche win for the third time in four games.
Rookie Matt Taormina scored his first NHL goal and assisted on Ilya Kovalchuk's third-period tally for the offensively challenged Devils, who have nine goals and three points in five games (1-3-1).
Anderson, who led NHL goaltenders in shots faced and saves last season, kept the Avalanche in the game in the one-sided first period that saw McLeod score late in a 4-minute power play and then get kicked out of the game 22 seconds later with a major boarding penalty.
During that 5-minute power play, Anderson stopped at least a half dozen great chances by the Devils and also got a little lucky.
A tip by Zach Parise slid just wide of the net and Dainius Zubrus failed to slam a puck into an open net after skating around Anderson in front.
Galiardi gave Colorado some breathing room in the second period, shoveling the rebound of Stewart's blast from the left circle into the top corner of the net for a 2-0 lead.
Taormina closed the gap to 2-1 3 minutes later with a power-play shot from straight on with Parise providing a screen in front.
Stewart, who led Colorado with 28 goals in their surprising run to the playoffs last year, got his second of the year early in the third period with a slap shot from the right faceoff circle that whizzed over the shoulder of Martin Brodeur, who made 19 saves.
Kovalchuk, who signed a high-publicized 15-year, $100 million contract in the offseason that has caused some salary cap and roster problems for New Jersey, got his second of the season with 11:11 to play.
Anderson maintained the lead the rest of the way, stopping a deflection by Patrik Elias with 5:45 to go and a point shot by Henrik Tallinder with 3:40 left. He also stopped blasts by Kovalchuk and Elias in the final minute when Brodeur was pulled for an extra skater.
Despite being outshot 17-3 in the first period, the Avalanche took a 1-0 when McLeod, who has had at least 100 minutes in penalties in his first three seasons, beat Brodeur to the short side with a shot from the left faceoff cirlce at 11:50.
Twenty-two seconds later, McLeod was given a 5-minute major and a game misconduct for boarding Taormina.
NOTES: In the first period of the last two games, the Devils have outshot Buffalo and Colorado by a combined 32-4 margin and been outscored 1-0. ... Jacob Josefson, the 19-year-old Swedish center who was the Devils' first-round draft pick in 2009, made his NHL debut. ... Colorado is 2-1 on its five-game trip with games left against the Islanders and Rangers. ... This was the only between the Devils and Avs this season.