Kane also had an assist while Alexander Burmistrov had a goal and an assist for the Jets, who had dropped their previous two. Mason made 22 stops in the win.
"Every game has such magnitude," said Mason. "Every game is a must win but chances are we're not going to win every one, so we take what's in front of us."
"It's not fun when you lose, so it wasn't fun," said Setoguchi. "We played a better game and scored, but our game isn't run and gun."
Minnesota struck first on a 3-on-1 break that saw Koivu fake the shot along the left wing before centering for the redirect from Setoguchi 7:50 in.
Winnipeg, though, struck back with 1:13 to play in the first as Blake Wheeler skated down the right wing and around the net before emerging on the other side and passing it over to Kane for the wide-open slam.
Kane scored three minutes into the second when he chipped the puck into the zone and from the left circle was able to get it over Backstrom, who had come out of the net to try and play the puck.
However, the Wild tied the game on the power play with 7:45 to play in the second as Setoguchi's small slap shot from the top of the right circle got by Mason.
The Wild made it a 3-2 game 1:03 into the third as a slap-pass by Jared Spurgeon at the right point went to the low left side where Koivu redirected it on net then slammed home his own rebound.
Winnipeg tied it two minutes later when Burmistrov was able to pick up a loose puck at the right circle off a bounce from the backboard and sneak it home.
Winnipeg hosts Boston on Friday...Minnesota plays in St. Louis on Saturday...Winnipeg won both meetings with Minnesota this season and has won four straight in the series.