Landeskog gave the Avalanche the lead, and Varlamov made it stand up as playoff-hopeful Colorado finished a 2-0-1 Eastern road trip.
The Rangers' once seemingly comfortable lead in the Eastern Conference is nearly gone with 11 games remaining. New York lost its second straight, and fifth in seven games, as surging Pittsburgh closes in. The Penguins, who beat New Jersey earlier Saturday to close within two points, can get even with the Rangers with a win Sunday in Philadelphia.
Colorado moved past idle Phoenix into seventh place in the West, but the Avalanche have played two more games than the Coyotes.
Defenseman Matt Hunwick got the Avalanche even in the first period after Mats Zuccarello gave the Rangers a 1-0 lead. Henrik Lundqvist returned from a two-game absence caused by the flu and made 18 saves in the loss.
Paul Stastny added an empty-net, power-play goal with 13.1 seconds left to make it 3-1.
New York also got back captain Ryan Callahan (bruised foot) and defenseman Michael Del Zotto (hip), but it wasn't enough. The Rangers have lost two straight at home, after winning six in a row there, and have three games remaining on a seven-game homestand.
They had their chances against Varlamov, but only got one puck past him. Brandon Dubinsky ripped a drive at him seven minutes into the third, but Varlamov went down in a split and made a lunging grab of the puck with his glove.
Later in the period, the Rangers had golden opportunities during a power play. Zuccarello hit the post with a drive, then got stuffed in front before Brad Richards was also denied from in-close. Another advantage in the final five minutes also proved fruitless as New York went 0 for 5 on the power play.
Despite dominating on the shot board with a 26-13 edge through 40 minutes, the Rangers trailed 2-1 going into the third.
Landeskog gave the Avalanche a 2-1 lead with his 21st goal of the season - tops among NHL rookies - at 9:51 of the second. Defenseman Ryan Wilson stepped into a shot, just above the left circle, that appeared to quickly tip off the stick of Rangers forward Ruslan Fedotenko and flutter toward the front of the net. Landeskog got his stick on it just in front of Lundqvist, and knocked it between the goalie's pads.
The Avalanche had control of the puck more in the second period than the first and deftly created several turnovers by the Rangers that either thwarted New York chances or set up opportunities for Colorado.
New York surely was frustrated to be in a 1-1 tie through the first period, despite having a 16-6 edge in shots.
Zuccarello, in his fourth game back with the Rangers since being recalled from the AHL last week, netted his first of the season 10:43 into the game. The 5-foot-7 winger worked his way to the front of the net, got his stick on Dan Girardi's shot from the right wing boards, and steered it between Varlamov's pads - even with Avalanche defenseman Erik Johnson leaning on his back.
It was Zuccarello's first goal since March 6, 2011 - a span of 16 NHL games - and the seventh of his career. The goal also was the first in three games scored by someone other than the potent line combination of Marian Gaborik, Richards, and Carl Hagelin, who had netted the past seven.
Notes: The Rangers were without C Artem Anisimov (shoulder) and D Steve Eminger. Defensemen Jeff Woywitka and Anton Stralman, along with hulking forward John Scott were also scratched. D Tim Erixon was recalled from Connecticut of the AHL on Saturday and was in the lineup. ... Colorado is 10-2 against the Rangers in the last 12 meetings. ... The Avalanche killed all five Rangers power plays, and the unit has allowed only one man-advantage goal in the past nine games (20 of 21).