Ryan Suter made it 3-1 at 14:43 with a one-timer from above the left circle.
"We played with energy and I thought we played the right way," Suter said. "We hadn't done that in about 10 games. The whole game, I thought the energy was there. Guys were working, and when we were winning games, that's how we were playing. That's how this team has to play."
Carter Soucy scored for the Wild at 16:02, but Cooper challenged for offside, and the goal was overturned after video review showed Kevin Fiala entered the Lightning zone ahead of the puck at 15:46.
"When we were going good, we had that pushback," Minnesota coach Bruce Boudreau said. "I thought we had it today, and my biggest fear was that one inch was going to cost us, like on that offside. I think if we could've [scored], they're a tired team and they might've folded the tent."
Kucherov scored his second goal of the night on the power play to make it 3-2 at 9:32 of the third period.
"We had some good looks and we got the big goal," Tampa Bay defenseman Victor Hedman said. "The power plays gave us some momentum, but we were passing up some opportunities to shoot."