Paul Stastny added two goals and Semyon Varlamov stopped 38 shots for Colorado, which is 8-0-1 in its last nine games and took over top spot in the Central Division ahead of the scuffling St. Louis Blues.
John Mitchell scored into an empty net for Colorado in the final minute to seal it.
The Blues, who have lost four straight, and Avalanche each have 111 points with two games left, but Colorado holds the tiebreaker.
Barrie's winner came with 7:30 left in regulation with the game tied at 2 and the teams playing 4-on-4. The 22-year-old defenseman moved in on a 2-on-1 rush, toe-dragged around a sprawling Alexander Edler and scored his 13th of the season, and fifth game-winner, past a helpless Jacob Markstrom.
Edler had a chance to tie it with just over five minutes to go, but Varlamov made a great glove save on the Canucks defenseman.
Markstrom was acquired as part of the Roberto Luongo trade with the Florida Panthers last month, but got into just one game in relief before Thursday as Eddie Lack started 19 straight in the Canucks' failed attempt to make the playoffs.
The Canucks were playing their first game since Trevor Linden was hired as the team's president of hockey operations Wednesday. Linden replaced president and general manager Mike Gillis, who was fired after Vancouver was officially eliminated from postseason contention for the first time in six years following a dismal 3-0 home loss to the Anaheim Ducks on Monday.
Stastny snapped a 1-1 tie with his second of the night and 25th of the season at 5:21 of the third, burying a rebound off Eric Johnson's shot from the faceoff circle.
Booth tied the game 1:39 later with his ninth of the season, roofing Dan Hamhuis' rebound off the end boards over Varlamov's shoulder from in tight.
Linden, who was shown on the video board before Thursday's opening faceoff and received a nice applause, didn't have to look far to see the kind of success former players can have in management. Avalanche executive vice president of hockey operations Joe Sakic has his team in a battle for top spot in the Central Division, while rookie coach Patrick Roy is in the running for the Jack Adams Trophy for a Colorado team that finished 29th last season.
Vancouver opened the scoring when captain Henrik Sedin tipped home his 11th of the season off a point shot from Hamhuis that found its way past Varlamov at 13:52.
Canucks coach John Tortorella said earlier this week that his young players would get a chance to play with three meaningless games left on the schedule, and a play by Vancouver rookie Niklas Jensen directly led to Colorado's tying goal with 1:21 to go in the period.
Stastny stripped Jensen of the puck at the Avalanche blue line and he raced in alone, snapping his 24th through Markstrom.
Canucks defenseman Yannick Weber was then denied on the doorstep by Varlamov with the teams playing 4-on-4 midway through period and the Colorado goaltender stood tall again with a nice save on Booth off the rush to keep things tied after 40 minutes.
NOTES: The Canucks will honor former coach and general manager Pat Quinn before Sunday's season finale against the Calgary Flames. ... Canucks defenseman Jason Garrison was a healthy scratch as Vancouver chose to dress rookie Frank Corrado. ... Avalanche forward Matt Duchene remained out with a knee injury.