Ryan Suter and Alex Galchenyuk each had two assists, and Alex Stalock made 23 saves for Minnesota (29-24-7), which trailed 1-0, 2-1, and 3-2. The Wild are three points behind the Calgary Flames and Arizona Coyotes for the second wild card into the Stanley Cup Playoffs from the Western Conference.
Ryan Nugent-Hopkins had two goals and an assist for Edmonton (32-22-7), which is second in the Pacific Division, one point behind the Vegas Golden Knights. Leon Draisaitl had a goal and an assist, Kailer Yamamoto had two assists, and Koskinen made 34 saves.
"Overall it wasn't a terrible game, but I think defensively we could have been a little sharper, a little quicker," Nugent-Hopkins said. "We gave them a little too much time, and they started rolling around [offensive zone]. When you have that much time, you start feeling good and getting some good opportunities."
Nugent-Hopkins scored on the first shot of the game, 50 seconds into the first period, to give Edmonton a 1-0 lead.
Kevin Fiala tied it 1-1 at 8:25, collecting a rebound, skating around Oilers defenseman William Lagesson and scoring on a backhand.
Nugent-Hopkins made it 2-1 at 13:04 on a shot from the high slot.
Carson Soucy tied it 2-2 at 5:48 of the second period on a long shot through a screen.
Draisaitl made it 3-2 at 8:01 when his centering pass hit the skate of Wild forward Jordan Greenway and bounced into the net.
"Their top line is a line that thrives off the rush," Wild defenseman Ryan Suter said. "We talked in between periods that we've got to eliminate the rush play with the Draisaitl line, and we did the second half of the game and came up with big blocks and great penalty kills. It resulted in a win."
Minnesota coach Dean Evason said, "I think the guys were very committed. We really committed to playing better defense. We were getting opportunities, so we were able to reenforce that between periods, that we're getting our chances, guys, but let's tighten up in all areas. The neutral zone was real good."
Spurgeon tied it 3-3 at 10:55 on a slap shot from the right face-off circle.
"It's a tough loss," Yamamoto said. "We need to tighten some things up. We just had a couple of tough breakdowns, things that we can tighten up pretty easily. I thought we played hard, we just didn't come out on top. These are things we can correct pretty easily. If we do that, we'll be a pretty deadly team."