Marcus Foligno scored two goals for Minnesota (5-1-2), which lost in regulation for the first time this season and had its four-game winning streak end. Joel Eriksson Ek had a goal and an assist, Kirill Kaprizov had two assists, and Filip Gustavsson made 16 saves.
"It was a little bit of a weird game," Wild coach John Hynes said. "(We) had some mistakes early, and then I felt we took the game over for a long period of time. ... There were some clear things that cost us the game tonight, but I did like the fight back, the pushback, Foligno scoring. There were lots of good components to the game, too."
Couturier made it 1-0 at 1:18 of the first period when he scored from in front off the rebound of Ristolainen's shot.
He scored on a similar play at 8:26 of the third period to make it 5-4.
"Especially when you're struggling for offense, that's where usually you find the back of the net," Couturier said. "Whether it's a lucky bounce off of you or second, third chances. And also, playing with playing with 'Meesh' (Michkov) and 'TK' (Konecny), my job (is) just got to control the center of the ice and create some space and go to the net and they'll make plays."
Seeler made it 2-0 at 3:51 of the first with a shot from between the top of the circles.
Brock Faber cut it to 2-1 at 14:54 of the second period, redirecting a shot by Kaprizov, and Foligno scored 33 seconds later off a rush from the right side to tie it 2-2.
"Our line, we played well all night," said Foligno, whose goal was his first of the season. "It's frustrating to lose in a good game. Our line the last three games has been really good. I'm more just happy we're getting offense going and it's nice to finally cash in."
Michkov gave the Flyers a 3-2 lead at 18:16. He scored on a shot from above the face-off circles during a delayed penalty with Ersson pulled for an extra skater.