The Jets (0-2-1), who were without captain Blake Wheeler after he entered the National Hockey League's COVID-19 protocol on Monday, played with 11 forwards and seven defenceman in the road trip finale.
The trio of Scheifele, Connor, and Andrew Copp combined for eight points in the loss - with Connor scoring a pair.
But while there were positives to build on, the loss was still tough to swallow.
"They got some lucky bounces and they ended up in our net," said Scheifele. "We played hard. It was a hard fought game. It was a rivalry game - us versus Minnesota. It was hard fought, fun hockey. Too bad it ended the way it did."
Connor got the Jets offence going with his second of the season just 3:48 into the first period. The Michigan product flew up the left wing and snapped a shot short side on Cam Talbot. Copp and Scheifele picked up assists on the goal.
That lead lasted less than a minute, as Mats Zuccarello pounced on a rebound that the Jets couldn't clear. That tied the game up 1-1 less than five minutes into the contest.
Connor put the Jets back in front with his second of the period at 8:17. Copp's play down the left wing boards went off a body and redirected into the slot where Connor was waiting. The three-time 30-goal scorer made no mistake, hammering a shot into the top right corner to put the Jets ahead 2-1.
Once again, Minnesota came back. This time it was Joel Eriksson Ek on a wrap around, taking advantage of the Wild's second power play of the period.
It put an end to an intense opening period that saw four minor penalties and a fight between Brenden Dillon and Foligno, and countless after the whistle shoving matches.
"I thought this was the first game this year that's had NHL intensity in it," said Jets head coach Paul Maurice. "The game's hard, lots of physicality, lots of good chances at both ends. Both teams were playing hard for their chances.
"We'll try to take some of it, but we'll clean up some things, obviously."
Winnipeg's power play got on the board 5:49 into the second to restore a one-goal lead. Paul Stastny fed Josh Morrissey in the right circle, and the Jets defenceman snapped it on goal quickly. The shot went off Talbot and in, giving the Jets their first goal on the man advantage this season.
The Jets power play finished 2-for-6 on the night.