Taylor Hall and Jordan Eberle each added a goal and two assists for the Oilers, who won for the second time in 11 games but will miss the playoffs for the seventh straight season. Nikolai Khabibulin stopped 38 shots.
"We wanted to come out and spoil their party, and we did," Hall said.
Mikko Koivu scored for the Wild, who could have gotten into the playoffs for the first time in five seasons with a win. Instead, Minnesota will enter Saturday's season finale at Colorado tied with Columbus for eighth place in the West, one point behind seventh-place Detroit.
Minnesota owns the tiebreaker and can still secure a postseason spot with a win.
If Detroit gets a point in its season finale, Columbus wins and Minnesota loses, the Wild would be eliminated.
"It's not the way we wanted it to happen, but we're still in a good place," defenseman Ryan Suter said. "We win and we're in the playoffs. That was our goal, to get in the playoffs, and once you get in the playoffs, you set another goal.
"We still control our own destiny here, and we have to go in there and play our best game of the season."
A lot was expected from the Wild this season after the club signed the top two available NHL free agents -- Suter and center Zach Parise -- to identical 13-year, $98 million contracts last July 4.
But the season started slowly, and fans began to grumble before the Wild got on a roll in March and briefly joined Vancouver atop the Northwest Division.
"I've got total confidence in our group to play that game tomorrow," coach Mike Yeo said. "This one hurts, but I've seen this group respond enough, I've seen our leadership enough to still feel confident."
A 4-8-1 mark in April -- including a 1-3 slide in their past four games, in which they were outscored 17-5 -- has moved the Wild to a dangerous position in the West.
With desperation settling in, Minnesota played one of its more complete games of the season Tuesday in a 2-1 win over Los Angeles.
Since then, Yeo continually reminded his players not to take anything for granted against the struggling Oilers.
The message didn't sink in.
Potter scored 27 seconds into the game when his one-timer from near the top of the right circle beat goalie Niklas Backstrom, who had lost his stick.
Hall made it 2-0 at 8:02 as he forced a turnover, picked up the loose puck at the right wall, cut into the right circle, and put a rising wrist shot through the legs of Suter and over the glove hand of Backstrom.
Minnesota's Matt Cullen missed on two quality chances on one shift late in the first period, and Edmonton countered at the other end, with Potter scoring his second of the period. He lifted a wrist shot over the glove of Backstrom, who was 17-0 against Edmonton at home.
Much to the delight of the capacity crowd, Backstrom, who allowed three goals on five shots, was pulled for Josh Harding, who hadn't played since Feb. 7 after he developed complications from the multiple sclerosis medication he has been taking. Harding finished with nine saves.
Yeo said Backstrom will start Saturday.
"It's tough to explain," center Kyle Brodziak said. "I don't think anyone really knows what happened. They came out and capitalized on every opportunity they had. They made the best of their opportunities, and we just gave them way too many."
Eberle scored 4 minutes into the second on a shot from below the right circle that beat Harding high on the stick side and scored again on a wrist shot midway through the period before Yakupov tipped in a pass from Hall to make it 6-0 at 13:55.
"Our goaltending was outstanding, and we used every one of our chances," Edmonton coach Ralph Krueger said.
Koivu scored midway through the third.
Minnesota's Zenon Konopka and Edmonton's Mike Brown fought 2 seconds into the game. ... Wild F Jason Pominville missed the game because of an upper body injury. He was elbowed in the head by the Kings' Dustin Brown on Tuesday and hasn't practiced since. Brown was given a two-game suspension by the NHL for the hit. ... Oilers D Ryan Whitney, who injured his leg Wednesday, missed the game. ... Wild LW Mike Rupp returned after sitting out four games with a lower body injury.
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ST. PAUL, MN - APRIL 26: Justin Falk #44 of the Minnesota Wild warms up prior to the game against the Edmonton...