Matt Calvert scored two goals and Francouz made 27 saves for the Avalanche (23-12-3), who are 1-3-0 in their past four games, blowing a third-period lead in each loss.
"They've all come in different ways," said Calvert, who has scored four goals in the past three games. "We've got to clean that up. If we do that, we're going to be a good hockey team. We made too many mistakes tonight."
Carson Soucy gave the Wild a 1-0 lead at 5:33 of the first period when his shot from the right circle made it through traffic.
Eric Staal made it 2-0 at 9:22, jamming in a loose puck in the crease after Fiala's one-timer trickled past Francouz.
"Sloppy start," Colorado coach Jared Bednar said. "We gave up a couple of goals and then I thought our game came around the second part of the first."
Calvert cut it to 2-1 at 18:24 on a redirection of Ian Cole's shot, and Gabriel Landeskog tied it 2-2 at 19:10 after he split Soucy and Brodin and scored with a backhand into an open net.
Joel Eriksson Ek appeared to score for the Wild at 8:37 of the second period, but Bednar challenged the play for offside, and the call was reversed after a video review.
Brad Hunt gave Minnesota a 3-2 lead with a power-play goal at 11:18 of the second. Minnesota had gone 2-for-30 with the man-advantage in its past 12 games before Hunt's goal.
Nathan MacKinnon tied it 3-3 at 19:12 after Cale Makar's shot from the right point redirected in off his left skate.
"It's kind of the way we've been for this whole first half of the year," Staal said. "To give up goals there in the last minute of both periods and still come back and continue to play and just stay even-keeled, it's a credit to a lot of guys in our room. Big win against a division team and we have to try to keep going."
Calvert gave Colorado a 4-3 lead 1:33 into the third period on a redirection of Pierre-Edouard Bellemare's shot.
"We had a one-goal lead with 10 minutes to go in the game," Landeskog said. "Again, it's a couple defensive breakdowns that give them the lead. It [stinks]."
