Jonathan Huberdeau scored two goals, Anton Stralman had two assists, and Sergei Bobrovsky made 36 saves for the Panthers (2-2-3), who are on a four-game point streak (1-0-3).
"We knew this was a good team," Bobrovsky said. "They're on a roll right now. We have to find a way to win these games."
Huberdeau gave Florida a 1-0 lead at 3:32 of the first period before Frank Vatrano scored shorthanded to make it 2-0 at 8:26.
Donskoi cut it to 2-1 at 10:57, but Huberdeau made it 3-1 at 14:57.
"I don't know what's going on right now," Huberdeau said. "We've got a lead and we cannot lose it every time in the second period. We've got to tighten up some things down defensively, especially our line. We got scored on twice. It can't happen. Obviously, we scored, but we can't get scored on."
Matt Nieto and Burakovsky scored 32 seconds apart for the Avalanche in the second period. Nieto scored on a redirection at 11:23 to make it 3-2, and Burakovsky tied it 3-3 on a wrist shot from the right circle at 11:55.
"I've been talking a lot and working a lot on shooting more," Burakovsky said. "The previous years I've always been looking up trying to find a pass instead of shooting. It's something I'm trying to get rid of. Obviously, I got a good release. It was nice to see it go in."
Florida took a 4-3 lead at 14:05 on Brett Connolly's goal, but Colorado needed only 16 seconds to tie the score again when Burakovsky made it 4-4 at 14:21.
Grubauer stopped Huberdeau on a breakaway 6:33 into the third period, then made a spectacular save on Aleksander Barkov on a power play at 9:41, getting his arm on a shot heading into an open net.
In overtime, Grubauer made two saves against defenseman Mike Matheson, first on a breakaway 52 seconds in and then with his glove on a wrist shot from the slot at 1:59.
"We needed [Grubauer] to make some saves, especially late in the game," Avalanche coach Jared Bednar said. "He was really good in the third. I didn't love his start. I thought he was fighting the puck a little bit, but when we needed him to make a save for us, he did it."