Charlie Coyle, Mikael Granlund and Zach Parise also scored for the Wild (24-16-3), who are 2-0-1 in their last three games after losing five of six.
"I think we're starting to turn a corner," Brodziak said. "It feels good to get these two points. Our team needed it. We're still in a real tough playoff race. We went through a definite lull where we weren't playing the way we should, but as of late, we are starting to find our game again.
"We still need to tighten some things up if we want to be a good playoff team, though."
Minnesota remained in sixth place in the Western Conference playoff race.
"We are in the process of trying to get better every night," said Wild goalie Niklas Backstrom, who only faced 19 shots. "There was a stretch where we had bad starts, and that cost us the game, but we came out very quickly in this game. It is always nice to get the early lead. You aren't chasing them and can play your own game.
"There are a lot of good things happening as of late and we just have to keep doing it night after night. You can't take any nights off right now."
Nail Yakupov, Magnus Paajarvi and Shawn Horcoff had goals for the Oilers (16-19-7), whose slim playoff hopes took another hit. Edmonton has lost six in a row on the heels of a five-game winning streak.
Oilers coach Ralph Krueger said he doesn't believe his team has given up on the season, despite the lack of positive results recently.
"When you are losing in a season like this, it is a bad spiral," Krueger said. "We were striving so hard to be a part of the playoff picture this year, and the disappointment has been large. It's taking us too long to recover. Our last three home games here have been extremely disappointing. But you have to refocus.
"We have six games remaining and we need to grab something positive from what is left here and be a team that shows character in this challenge. We will work hard to make these games valuable."
The Oilers again fell behind early as the Wild scored just over a minute in. Brodziak sent a knuckler of a shot on goalie Devan Dubnyk and got in front to knock in the rebound.
"When you get scored on early every period it's no fun," Oilers forward Taylor Hall said. "It's a painful way to lose. I've been here three years and I'm tired of this. We've got to start winning games just to get a better mood in here.
"Everyone cares in here, but it's fair for you guys to judge us in this situation."
Edmonton tied it 1-1 with 15 minutes gone in the opening period as Taylor Hall's shot got caught up in the equipment of Wild defender Brett Clark before falling in front of the net. Yakupov swooped in and scored the 11th goal of his rookie season, beating Backstrom.
The Wild regained the lead three minutes into the second period as a point shot from Stoner clipped Coyle in front and caromed past a screened Dubnyk to make it 2-1.
Minnesota went up 3-1 three-and-a-half minutes later as a rebound off the boars was tipped out front to Granlund, who slammed the puck into a wide-open net as Dubnyk went the other way.
It took just 20 seconds into the third for the Wild to go up 4-1 with a power-play goal. Parise slid the puck through the crease and went behind the net untouched. He still had time to backhand in his initial shot for his 16th goal.
The Wild went ahead 5-1 five minutes into the third as Jason Pominville sent Brodziak in on a breakaway. Brodziak hit the post with his shot, but Dubnyk put the puck into his own net with his backward momentum.
Edmonton got a goal back at the 10-minute mark as Hall sneaked a pass through the crease to Horcoff, who was stationed in front, and tapped home his seventh goal.
The Oilers made it a bit more interesting less than three minutes later as Yakupov worked in-between traffic before ringing a shot off the post. Paajarvi swooped in to hammer the rebound past Backstrom to make it 5-3.
NOTES: It was the third of four meetings between the teams this season. The Wild have won them all. Minnesota has won four straight against the Oilers and 14 of 17. Minnesota has also won seven of eight in Edmonton. ... It was the Oilers' first game since Monday's announcement that general manager Steve Tambellini was fired, with former head coach Craig MacTavish coming in as his replacement. ... Edmonton called up forwards Anton Lander and Teemu Hartikainen from Oklahoma City of the AHL and inserted both players into the lineup. Ryan Jones and Jarret Smithson came out.