Mason Raymond also scored and Roberto Luongo stopped 38 shots.
Before the game, the Canucks clinched the division title courtesy of Calgary's loss to Dallas. They also closed the gap on St. Louis in the race for the top seed in the conference, climbing to within two points of the Blues.
Gabriel Landeskog and Jamie McGinn scored for the Avalanche, who have earned at least a point in nine of their last 11 games to remain in the thick of the playoff chase.
On the game-winning play, Jannik Hansen outraced defenseman Tyson Barrie to the puck in the corner and fed a wide open Higgins, who lined it past Semyon Varlamov.
Luongo shined in net all night as the Canucks improved to 5-0 against Colorado this season.
Vancouver was without Daniel Sedin for a second straight game with concussion-like symptoms.
The star forward was hurt when Chicago Blackhawks defenseman Duncan Keith lunged at Sedin in the neutral zone in the first period of Chicago's 2-1 overtime victory last Wednesday, catching Sedin in the head with his elbow. Keith was suspended for five games by the NHL.
The Avalanche begin a pivotal road stretch as they travel to San Jose, Vancouver and Calgary over a five-day stretch.
Matt Duchene had a solid scoring chance late in the third period as he gobbled up a rebound and glided in on Luongo, only to fan on the shot.
Varlamov showed he's an adept passer as he set up the Avalanche's second goal. With the Canucks making a line change, the young Russian goalie sent a long outlet pass to Paul Stastny, who dished it over to McGinn and he took it from there, scoring his 20th goal of the season.
When Varlamov's name was announced for the assist, the capacity crowd roared.
Soon after, Vancouver came roaring back from the 2-0 deficit as the Canucks figured out a little flaw in Varlamov's game - the high shot.
Raymond and Higgins both went high to beat Varlamov.
Colorado's penalty kill unit was in fine working order early in the game, as the team weathered one power play after another.
Near the end of the first period, David Jones was called for a double-minor after his stick hit Kevin Bieksa in the face and drew blood. The Avs bottled up the Canucks for the rest of the period and then the opening 1:34 to start the second period.
This after surviving a 5-on-3 opportunity by Vancouver for more than a minute in the middle of the first period. A few kick saves from Varlamov and three blocked shots with the body by Ryan O'Byrne helped the Avs maintain their lead.
Colorado scored early in the first period on a rare power-play opportunity when Hejduk's shot ricocheted off the pads of Luongo and a hustling Landeskog slapped it in. Colorado averages just 2.7 power plays a game, the fewest in the league.
Landeskog is making a strong case for the Calder Trophy, which is awarded to the top rookie. He leads all first-year players in goals, shots and takeaways.
Chuck Kobasew made his return to the lineup after sitting out nearly a month with a sore back. It didn't take long for him to get reacquainted with the action as he decked Aaron Rome in retaliation for the defenseman's earlier hit on Tyson Barrie. That led to a brief skirmish between Kobasew and a couple of Canucks players.
Luongo came up big often in the opening period as he turned back 16 of 17 shots, including a glove save on a backhanded attempt from Ryan O'Reilly.
(Copyright 2012 The Associated Press. All Rights Reserved)