Jakob Silfverberg had two goals and an assist, Rickard Rakell had three assists, and Ryan Miller made 25 saves for the Ducks (28-32-9), who had won two straight.
"We were lucky we connected a few times," Rakell said of Silfverberg. "That just gives us confidence going forward. It's something we can build on."
Victor Rask stole the puck from Ducks defenseman Jacob Larsson in the neutral zone and cut toward the slot before shooting under Miller's arm to tie it 3-3 at 1:49 of the third period.
"Guys just talked about how much we needed the two points and that we've got to bear down and go at them," Fiala said. "With this group, over the last long stretch here, we certainly believe in ourselves to be able to do it."
Miller saved a breakaway by Zach Parise at 3:08 and a shot from Mikko Koivu from in close at 7:19 before Galchenyuk gave Minnesota a 4-3 lead at 14:20 following a giveaway by Nicolas Deslauriers.
"There were a lot of bouncing pucks," Galchenyuk said. "It bounced right on my stick."
However, Christian Djoos scored his first of the season with Miller pulled for the extra skater to tie it 4-4 with 1:42 remaining in regulation.
Silfverberg gave Anaheim a 1-0 lead at 11:18 of the first period. Rakell took possession of the puck along the boards and passed it to Silfverberg for a shot that beat Dubnyk glove side from the left circle.
"We bring out the best in each other," Silfverberg said of Rakell.
Mats Zuccarello scored his first goal since Feb. 1 (18 games) to tie it 1-1 at 12:02, shooting into an open net after his pass for Galchenyuk on a rush was blocked by Manson.
Ducks coach Dallas Eakins challenged the play for goaltender interference, but the call was confirmed after a video review.
On the ensuing penalty for delay of game, Fiala scored short side from the right circle to give the Wild a 2-1 lead at 13:33.
Danton Heinen tied it 2-2 at 11:52 of the second period. He carried puck along the goal line from the corner before Wild defenseman Jonas Brodin knocked it loose between Dubnyk and the right goal post.
Silfverberg scored his 20th of the season after Rakell stripped Joel Eriksson Ek in the neutral zone to make it 3-2 at 14:55.
The Ducks went 0-for-4 on the power play. The Wild were 2-for-3.
"We'd like to improve on that," Silfverberg said. "If we get one or two there, it's obviously a different game."
