Carson Soucy flattened Jets forward Nikolaj Ehlers after the Jet winger interfered with him in in the Wild end 7:46 into the third period.
The decision proved to be a poor one for Soucy as both took a seat in the penalty box and allowed the more skilled Jet forwards more room to freewheel.
Kyle Connor converted a Patrik Laine feed as both forwards converged on a hapless Devan Dubnyk before Connor fired a wrister past him.
Laine was late entering the zone as Jet defenceman Neal Pionk gained the blue line and curled off toward the boards before finding a streaking Laine, who slid the disc to Connor who promptly fired the game winning tally.
Jack Roslovic scored just 28 seconds later after he knocked a Josh Morrissey rebound past Dubnyk to give the Jets a two-goal cushion.
Patrik Laine had a four point night with two goals and two assists to pace the Winnipeg offence.
The Jets winger registered his 10th point on the season in his fourth game back after missing all but one day of training camp before ending his contract holdout and signing a two-year bridge deal with the Jets.
Blake Wheeler scored and added an assist and Mark Scheifele had a pair of assists for Winnipeg.
Connor Hellebuyck won his second straight game, stopping 35 shots for the Jets.
Devan Dubnyk turned aside 27 shots in the loss for the Wild, who have yet to win this season.
Minnesota scored first 13:08 into the opening period when Jets defenceman Ville Heinola turned the puck over right in front of the Jets goal.
Hartman spun the Jet defenceman around from behind, sending him to the ice before pouncing on the loose puck. Hartman wasted no time, beating Jets goaltender Connor Hellebuyck with a low shot to make it 1-0 Wild.
The goal was his first of the season.
Blake Wheeler tied the contest 1:37 into the second period after Mark Scheifele beat Jared Spurgeon in a battle behind the net before feeding the Jet captain a backhand pass which he promptly snapped past Wild goaltender Devan Dubnyk to make it 1-1.
The goal was his third of the season and marked the Jet captain's 700th NHL point.
Winnipeg fans gave him a standing ovation.
Patrik Laine gave the Jets the lead 9:10 into the second, one timing a cross ice feed past Dubnyk from the top of the right face off circle to make it 2-1
The goal was his second of the season and 8th point of the season.
Brad Hunt tied the game 3:09 into the third after moving in from the point before firing a high wrister through a screened Connor Hellebuyck to make it 3-3.
With both teams playing 4-on-4, the Jets took advantage of the room when Patrik Laine skated in before sliding the disc to Connor who fired a low shot past Dubnyk to make it 3-2.
Jack Roslovic made it 4-2 just 28 seconds later.
Patrik Laine added an empty netter with 2:19 left to make it 5-2 Jets.
SHOTS-MINNESOTA 40 WINNIPEG 32