It was Winnipeg's first victory in a postseason game since the NHL returned to the city for the 2011-12 season.
Mark Scheifele and Patrik Laine scored earlier for the Jets.
Minnesota's Matt Cullen and Zach Parise scored in a little more than a two-minute span early in the third period.
Four of the game's goals came in the third period.
Winnipeg goalie Connor Hellebuyck racked up 18 saves. Wild goalie Devin Dubnyk made 37 stops.
The Jets posted an NHL-best 32-7-2 home record during the regular season. Winnipeg is in the playoffs for just the second time in seven seasons since the NHL returned to the city, and it was swept out of the postseason the first time in 2015.
The Wild are in the postseason for the sixth consecutive season.
The Jets scored the game's final two goals, with Morrow's shot from the point making it to the net.
With a blast into the top corner, the 19-year-old Laine tied the game 53 seconds after Parise's goal gave the Wild its first lead.
So it took less than five minutes of the third period for the teams to combine for three goals.
Scheifele converted on a power play with 2:23 remaining in the second period.
Blake Wheeler and Dustin Byfuglien were credited with assists. Minnesota's Nate Prosser was in the penalty box for holding on Scheifele's goal.
It was a scoreless first period, though the Jets held a 13-4 edge in shots on goal.
Both teams killed off second-period penalties before Winnipeg struck on its second power play.
The Winnipeg penalty was a roughing infraction on Byfuglien, who was active with a team-high three shots on goal through two periods.
Winnipeg lost forward Mathieu Perreault in the third period to an upper-body injury.
The Jets entered the postseason with victories in 11 of their final 12 regular-season games.
--Field Level Media
Minnesota | Winnipeg | |
Matt Cullen 1 | Points | Patrik Laine 1 |
Matt Cullen 1 | Goals | Patrik Laine 1 |
Mikael Granlund 1 | Assists | Dustin Byfuglien 1 |
N/A | Power Play Goals | Mark Scheifele 1 |
N/A | Short Handed Goals | N/A |
Devan Dubnyk .925 | Save Percentage | Connor Hellebuyck .900 |
Devan Dubnyk 37 | Saves | Connor Hellebuyck 18 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Minnesota | 20 | 2 | 0-1 | 1-2 | 4 | 26 |
Winnipeg | 40 | 3 | 1-2 | 1-1 | 2 | 31 |