Rakell scored in front to give the Ducks a 1-0 lead at 1:18 of the first period.
The Wild have allowed the first goal in every game this season.
"Obviously, we'd like to start with a lead, but again we love our resilience," Minnesota coach Dean Evason said. "I think it's just who they are, right? … We were happy with that end of it."
With the teams at 4-on-4, Duhaime deflected a shot by Nico Sturm at 7:11 to tie it 1-1. Jared Spurgeon put Minnesota ahead 2-1 at 9:05.
Troy Terry scored on a deflection to tie it 2-2 at 11:11.
Rakell put the Ducks ahead 3-2 at 1:49 of the second period with a power-play goal.
"This game could have gone either way, so it's a little bit disappointing because I thought everyone played a really hard game, gave them a good fight," Rakell said. "It could have been going either way tonight."
Jon Merrill one-timed a pass from Marcus Foligno to tie it 3-3 at 11:56.
"We're having a lot of fun as a group right now, it's a good group of guys," said Merrill, a defenseman who signed with Minnesota as a free agent July 29. "… Winning helps all that, and we're just looking to keep this streak going.
"There's a lot of belief in our group, so I think whether we're down or we're up, the message is pretty much the same. Once we get to our game, which I think is the offensive zone, getting in on the forecheck, kind of seeking their D and our big, heavy forwards turning pucks over, and we kind of get the spin cycle going and we get things moving around in the offensive zone, we've had some success that way early on. So we know that once we get to that game, we can play with anyone in the League."
NOTES: The Wild started 6-0-0 in 2006-07, 5-0-0 in 2007-08 and 4-0-0 in 2008-09. … Talbot has won each of Minnesota's four games. ... Foligno extended his point streak to four games (four points; two goals, two assists). … Merrill scored his first goal since Feb. 8, 2020, for the Vegas Golden Knights. … Terry has scored six points (two goals, four assists) in his past four games. ... The Ducks went 1-2-1 on their four-game road trip. ... Anaheim defenseman Kevin Shattenkirk had a five-game point streak end.