Dubois gave Winnipeg a 1-0 lead on a breakaway at 18:34 of the first period after a pass from Blake Wheeler.
Eyessimont's first NHL goal made it 2-0 at 6:44 of the second period. After Kochetkov stopped David Gustafsson on a breakaway, the 26-year-old forward finished on the rebound from the right hash marks.
"It was a one-knee drop on my forehand," Eyssimont said. "I had quite a bit of net to shoot at, and there were some bodies flying around the net. I just had to tap it in."
Morrissey gave the Jets a 3-0 lead at 3:56 of the third period when he picked up a drop pass from Wheeler inside the offensive zone, drove the net and scored on a wrist shot.
"We were really good, we didn't give them much," Rittich said. "That's what we've talked about since [training] camp: to be resilient, to work hard, to play 60 minutes, to do right things at right times, which we're doing."
Winnipeg coach Rick Bowness gave his team credit for working through Carolina's late comeback.
"We'll focus on the really good 55 minutes that we played, and we will figure out what we could have done better in the last couple," Bowness said. "Again, give our guys credit. Perseverance. You stay and fight."