Corey Perry scored for the Ducks, who lost on the road for the first time in seven games. Anaheim remains three points behind Phoenix in the Pacific Division, and is tied with three other teams for sixth place in the Western Conference standings. The Wild are 10th, one point back.
"It was a huge win versus a team that's ahead of is in the standings and that we're chasing," said Wild coach Todd Richards.
Anaheim goaltender Curtis McElhinney struggled in his second straight start since the Ducks placed starting goaltender Jonas Hiller on injured reserve with a recurrence of fatigue and lightheadedness. McElhinney allowed seven goals Wednesday to Washington, calling his performance "disheartening."
He said the phrase is again appropriate.
"There need to be some saves made at critical points in the game, and they aren't happening right now. I need to find a way to get it done."
Coach Randy Carlyle was noncommittal about Saturday's goaltender in St. Louis, saying "You'll have to be at warmup." Timo Pielmeier, who backed up McElhinney on Friday, has yet to play in an NHL game. Veteran Ray Emery, who signed a one-year deal with Anaheim on Monday, is on a conditioning stint with the Ducks AHL team in Syracuse.
Carlyle said McElhinney's positioning and a "couple of other things" could have been better, but he was critical of the rest of the Ducks for not making the goaltender's job easier. "We gave up too many odd-man rushes, Corey Perry falls down and they get a four-on-two," he said.
Less than two minutes after Nystrom scored on that rush for a 3-1 Minnesota lead, Anaheim defenseman Cam Fowler failed to clear Brodziak from the crease and the Wild center was able to swat in a rebound of Cal Clutterbuck's shot.
"We were in situations where we're not supposed to do that, and those usually end up in our net. We just have to keep it more simple and play smarter," defenseman Toni Lydman said.
Minnesota goaltender Niklas Backstrom continued his sensational month, stopping 29 shots. He has allowed just eight goals in six February starts.
His lone misstep came early in the first period when Perry tipped a harmless looking shot from Bobby Ryan off Backstrom's left leg and into the net. However, Koivu tied it 91 seconds later, fighting off Lubomir Visnovsky as he came out from the end boards and stuffing the puck past McElhinney on his second try.
Koivu left the game with a hand injury after the first period and did not return. He is expected to be re-evaluated Saturday.
"To lose a guy like we did during the game, everyone's got to step up a little bit and we did. He's a huge player on our team and it's a big void when he's not in there," Brodziak said.
Cullen scored short-handed late in the first period for a 2-1 Wild lead.
During a four-second 5-on-3 for the Ducks, Brodziak intercepted a pass from Visnovsky in the Wild end, took a stride and saw Cullen sprinting out of the penalty box behind the defense. Cullen went in alone and beat McElhinney high on the stick side. It was his fourth short-handed goal of the season to tie him with Boston's Brad Marchand and Frans Nielsen of the New York Islanders for the league lead.
Havlat scored a power-play goal in the third period.
Perry has 14 points during his current seven-game point streak. ... D Marek Zidlicky returned to the Minnesota lineup after missing 21 games with a separated shoulder. ... Anaheim fell to 9-2-2 against the Northwest Division. ... Minnesota is scheduled to practice outdoors Saturday for the first time in team history.