Barrie also added two assists and Calvin Pickard had 25 saves for Colorado. The Avalanche are 9-21-2 at Pepsi Center and had lost 12 of their past 14 overall before rallying to beat the Hurricanes.
Colorado has scored an NHL-low 126 goals this season and has been shut out 11 times. It took nearly two full periods to finally break through against the Hurricanes.
"I liked the way were playing tonight and then they go through almost two periods and not have scored," Colorado coach Jared Bednar said. "I was starting to wonder again, but we had good energy, our guys were positive on the bench."
Justin Faulk scored for the Hurricanes and Eddie Lack stopped 22 shots. Carolina is 2-5-3 in its past 10 games and is tied with New Jersey for last place in the Metropolitan Division.
The loss spoiled the first NHL game in Colorado for Hurricanes defenseman Jaccob Slavin. He grew up outside of Denver and played for Colorado College before breaking in with Carolina last season.
"It's really special growing up here watching the Avs play and growing up 30 minutes from here," he said. "It's definitely really special to come back and play here at the Pepsi Center."
The Avalanche had not scored more than a goal in their previous four games before Landeskog and Barrie struck less than a minute apart.
Landeskog tied the game when he tipped Barrie's shot past Lack at 18:18 of the second period. Landeskog's goal, his 15th, was Colorado's first power-play goal in 24 chances over a span of 11 games.
Barrie quickly gave the Avalanche the lead with his sixth of the season.
"Obviously we were having some difficulties, two goals in four games and we were getting frustrated," said Barrie, whose three points were a season high for him. "It didn't seem like we were ever going to score, so it was nice to get one and then get one right after."
Nathan MacKinnon scored his 13th goal of the season into the empty net with 50 seconds left. It was his first goal in 16 games.
The Hurricanes took a 1-0 lead when Faulk scored at 6:08 of the first period. Carolina appeared to go up 2-0 on Teuvo Teravainen's shot from the point midway through the second, but his goal was disallowed for goaltender interference. The Hurricanes challenged the call, but it held up under review.
"There was no real interference other than our guy's stick rolled up high," Carolina coach Bill Peters said. "When we first got to look at it on the bench we didn't have that angle of the stick. We knew there was no contact in the blue paint, but when we got the later look obviously the stick to the throat area was impactful."
Carolina LW Phillip Di Giuseppe was scratched with an upper-body injury. Di Giuseppe was hurt during the second period of the Hurricanes' 2-1 win at Arizona on Sunday night. With Di Giuseppe out the Hurricanes, played with 11 forwards and seven defensemen. ... Landeskog's goal ended a scoring drought of 98 minutes, 18 seconds for the Avalanche. Colorado's Matt Duchene leads the team with 16 goals, but hasn't scored in 21 games. He had 30 goals last season. ... Hurricanes LW Brock McGinn missed his fourth straight game with an upper-body injury.