The stirring victory came two nights after the Canucks blew a two-goal lead in the third period in a loss to Vegas.
“A lot of times we hold the sticks too tight when we do get the lead,” Vancouver coach Bruce Boudreau said. “We need a couple of these."
Colorado's Mikko Rantanen scored twice in the first period. He left after taking a high stick to the face in the second and then returned in the third.
Avs coach Jared Bednar said he was bleeding from the nose and gave him kudos for returning.
Makar also scored for the Avs, who lost Evan Rodrigues to a lower-body injury late in the first. Francouz stopped 22 shots.
Colorado had won three straight. After going 32-5-4 at home last regular season and winning the Stanley Cup in June, the Avalanche fell to 4-3-1 at Ball Arena.
There was a brief delay with 1:03 to go when fans, already upset with the Makar interference call, threw debris on the ice after Colorado took its fifth penalty.
“You can't have the mentality of worrying about the refs because it's just a tough game to call, probably the toughest in sports,” Makar said.
With the game tied at 2, Tyler Myers jammed his stick into Rantanen's face in the second, earning a double minor. The Avs ended up with a 5-on-3, and Makar took a feed from Nathan MacKinnon (NHL-best 22nd assist) and blistered a shot from the left circle with 4:01 left.
But Dries knocked in a loose puck early in the third period to tie it.
It marked the seventh straight game the league's best power-play unit scored.
Colorado, coming off a 3-0 trip, got off to a sloppy start.
A turnover behind the net led to Elias Pettersson finding an open Mikheyev in the slot, and his wrist shot 21 seconds in made it 1-0. It was the fastest road goal for Vancouver since Henrik Sedin scored 10 seconds in at Colorado on Oct. 24, 2014.
Artturi Lehkonen assisted on both Rantanen goals, extending extend his career-best point streak to eight games. ... The Avs also recalled F Jayson Megna. ... Avs D Samuel Girard (lower body) returned after a six-game absence. ... Canucks F Tanner Pearson (hand) skated in the morning but missed his sixth straight game. ... Injured Avs captain Gabriel Landeskog (knee surgery) turned 30.
Canucks: Visit Vegas on Saturday night.
Avalanche: At Nashville on Friday.
AP NHL: https://apnews.com/hub/nhl and https://twitter.com/AP_Sports
Mike Cranston, The Associated Press