Anthony Cirelli had a goal and an assist, and Andrei Vasilevskiy made 16 saves for the Lightning (24-13-1), who went 0-for-4 on the power play and have been outscored 9-3 in their past two games.
Tampa Bay played without defenseman Victor Hedman, who was with his wife for the birth of their child.
"It's a complete lack of discipline, it was embarrassing what we did," Lightning coach Jon Cooper said. "How many 5-on-3s you're up against in one season, you count on maybe one hand, and we give them two minute-and-a-half 5-on-3s in one game. Unacceptable. That shouldn't happen. And good on Winnipeg."
The Jets went 2-for-6 on the power play, which each goal coming on a two-man advantage.
"Penalties have been an issue for our team up and down the lineup," Lightning defenseman Ian Cole said. "Just too many. I am certainly a part of that. I can be way better.
"Fact of the matter is, I think our PK has been really good the past little bit. I mean, maybe not the past couple games, but the past two months or so. But what it does is it takes our best players off the ice."
Cirelli gave Tampa Bay a 1-0 lead at 11:48 of the first period, scoring on a rebound from the edge of the left circle following a shot from Steven Stamkos.
Dubois tied it 1-1 at 19:21 during a 4-on-4 situation when he redirected Connor's shot from the left circle.
"We knew it could be a one-goal game, a 1-0 game," Dubois said. "It's a good measuring stick for us … a big team win for us. The penalty kill did a fantastic job in killing all those penalties. This game we could look at it, and it could be a game we remember at the end of the season for us."
Nikita Kucherov put the Lightning back in front 2-1 at 1:35 of the second period with a wrist shot through traffic.
Dubois tied it 2-2 with his second goal of the game on a power play at 7:05, redirecting Morrissey's one-timer from the high slot.
