The Avalanche rallied with two goals in the last 10:14 of regulation to send the game to overtime, with Landeskog scoring the equalizer with 4:47 left.
Each of Colorado's goals went directly off a Columbus player in front of befuddled goalie Sergei Bobrovsky (25 saves).
"We were the best shooters in the game tonight vs. our own guy," Blue Jackets coach Todd Richards said. "To have three pucks go in off you is deflating. It takes away a lot of the good things we did in the game."
Colorado (48-21-6) moved three points ahead of the third-place Chicago Blackhawks in the Central Division, and Columbus (38-30-7) maintained the second Eastern Conference wild-card spot for the Stanley Cup Playoffs.
"I don't look at it as a loss," Richards said. "We still got a point. You'd like to get two points, but we did get one. That might be a huge point. Who knows? We'll find out in a couple of weeks."
Columbus (38-30-7) led 2-0 with 10:14 left in the third on second-period goals by Cam Atkinson and Blake Comeau until its fortunes turned, capped by Landeskog's game-winning goal with the Avalanche on their first power play.
Landeskog drew a holding penalty on Artem Anisimov with 2:21 left in overtime. With 32.2 seconds remaining he tried to send a pass through the crease to Nathan MacKinnon from the right side. Instead, the puck went off Columbus defenseman Jack Johnson's leg and into the net.
"It was a lucky play," Landeskog said. "I was going to Nate there. Johnson went down, and it went off his shin pad, I think, and went in far side. That's as lucky as they come."
Landeskog's 26th goal of the season was an exclamation point on a stunning comeback for the Avalanche (48-21-6) and continued their dominance with the man advantage against the Blue Jackets. In a 5-3 win against the Blue Jackets on Dec. 31, Colorado scored on all three of its power-play chances.
This time, the Blue Jackets appeared to be on the verge of killing off the man advantage, thanks to the strong work of Brandon Dubinsky, but the Avalanche were able to set up one last time to earn the victory.
"We had a bad 10-minute stretch in the second," Colorado forward Paul Stastny said. "We were down 2-0, but it didn't feel like that. We thought we'd get one goal and get back into it."
That's what happened.
Bobrovsky lost the shutout with 10:14 left in the third when Colorado center Brad Malone was credited with his second goal of the season after his centering pass from the left corner caromed off Columbus left wing Nick Foligno, the only player standing in front of Bobrovsky.
Landeskog scored 5:17 off an assist by MacKinnon to tie the game. The rookie, moved to the top line when Avalanche leading scorer Matt Duchene sustained a knee injury Saturday that will keep him out up to four weeks, took a pass from low in the right circle, then fed a pinpoint pass to the goalmouth for Landeskog, whose redirect went in off Blue Jackets defenseman Dalton Prout.
"Those are the bounces. That's sports," Johnson said. "Usually, those even out in the course of an 82-game season. Hopefully, we'll get a couple of those down the stretch."
The Blue Jackets had some help of their own in breaking the scoreless tie in the second period. Each goal was created off a turnover.
Dubinsky set up the first goal by swiping the puck from Colorado defenseman Nate Guenin while they battled for possession behind the net. Dubinsky took control and fed Atkinson in the slot for a one-timer at 10:52 past Colorado goalie Semyon Varlamov (24 saves). It was Atkinson's 20th goal.
Guenin, a former Ohio State player, was the unlucky victim five minutes later on a play that started with Columbus defenseman Fedor Tyutin flipping the puck out of his own zone to center ice. Avalanche defenseman Tyson Barrie whiffed at his attempt to corral the puck, which then bounced over Guenin's stick. Comeau poked the puck from him and dished it left to Foligno. Comeau continued toward the net and was in perfect position to receive the return pass from Foligno for the quick shot.
"It took some crazy bounces, and I was able to get to the puck first and pass it over to Nick," Comeau said. "He made a great play to me at the backdoor."
Colorado coach Patrick Roy was pleased with the response after allowing the two goals.
"I liked the way we played in the first and third," he said. "I'd liked to cut down on the turnovers in the second. Those things are part of the game. We bounced back and showed a lot of character."
The Blue Jackets play at the Philadelphia Flyers on Thursday; the Avalanche host the New York Rangers the same night.