Boone Jenner scored into an empty net with three seconds left for the 3-1 final.
Korpisalo made 14 saves in the third period.
"The guys did a good job letting me see the puck, blocking shots," he said.
Connor gave the Jets a 1-0 lead on the power play at 19:58 of the first period when he scored from the right dot off a pass from Blake Wheeler for his 25th goal of the season.
"Give their goalie credit, he played a good game," Winnipeg coach Rick Bowness said. "But the other point is there were times we had some Grade A's we missed the net on, and too many times he saw the puck and trapped the puck, so the greasy goals weren't going to be there in that period. But we played a really good period."
The Jets had a 12-2 advantage in shots in the first period, tying the Blue Jackets' season-low set in the first period against the San Jose Sharks on Jan. 21.
"It wasn't the start we wanted, but our goalie kept us in it," Columbus coach Brad Larsen said. "There was nothing going on in that period. All the best teams, you watch them, and they have periods like that. They do. But one goal is a lot different than three. You get some momentum, and you're one shot away."
Laine tied it 1-1 at 10:00 of the second period on a power play when his shot from the left circle went to the upper-right corner for his 15th goal but first in eight games. He has three goals in two games this season against his former team.
"Our power play was pretty good tonight," Laine said. "We probably could have had a couple more."