Oliver Ekman-Larsson scored twice, and Darcy Kuemper made 25 saves for Arizona (37-33-8), which has lost six of its past seven games.
With the Coyotes' loss, the St. Louis Blues and Vegas Golden Knights each clinched a berth in the playoffs.
"You can figure out the math," Coyotes coach Rick Tocchet said. "We needed two (points), but we got one to stay alive. Four games are left, and we need every point and every bit of help from other teams. We didn't give up. Oliver did a great job of getting us back in the game."
Ekman-Larsson cut it to 2-1 at 12:00 of the third period on a shot from the left point before tying it at 19:09 on a shot from the left circle that beat Grubauer five-hole after Kuemper had gone to the bench for the extra skater.
"We got back in the game by competing," Ekman-Larsson said. "I got the goals that gave us a chance. We said that we have to keep working hard and that's all we can do and maybe we'll get back in the game. That's what we did, and I think that could be a huge point at the end of the day. So we'll take that one and move forward and hopefully get in the playoffs. It's going to be a little bit harder now."
The Coyotes outshot the Avalanche 20-7 in the third period.
"They threw more pucks at the net and that's their system and it worked for them in the third period, but I didn't think we gave up a lot," Bednar said. "We have to find a way to continue to push and create offensively in the third period when we're leading."
MacKinnon gave the Avalanche a 1-0 lead at 13:37 of the second period, taking a pass from Gabriel Landeskog that went through the legs of Coyotes defenseman Jason Demers before chipping a shot over a sprawling Kuemper.
Landeskog was playing his first game since March 7 (nine games) because of an upper-body injury.
Brassard made it 2-0 at 19:26, getting to a loose puck at the right hash marks with a delayed penalty against Coyotes forward Richard Panik.
"They have a good penalty kill, so we were ready for pressure," MacKinnon said. "They're first in the League on the penalty kill (entering the game) and we did a great job on the power play. It's nice to have [Landeskog] back. He makes really good plays, he kind of settles things down. Obviously, great to have him back."
MacKinnon scored the only goal of the shootout in the first round.
"It was a little bit closer than we wanted it to be," Grubauer said. "We need to stay on the gas a little bit in the last period. It's quite a bit loud in here going into the last period with a lead and we were giving up a lot of shots. But on the other side, we're doing a great job defending, too."
