Kyle Connor had a goal and an assist for the Jets (35-30-11), who have lost six of eight including 6-1 at the Florida Panthers on Friday. Connor Hellebuyck allowed seven goals on 32 shots before being replaced by Eric Comrie (five saves).
Winnipeg fell to 10 points behind the Dallas Stars for the second wild card into the Stanley Cup Playoffs from the Western Conference.
"I don't know whether it's tired, lack of motivation from some guys in the room. We're definitely a frustrated group here," Connor said. "Yeah there is a slight chance, but it's a chance to make the playoffs. ... Two critical games and we put up that kind of effort."
Paul pulled Tampa Bay within 4-3 at 13:56 of the second period, and Kucherov tied it 4-4 at 18:03, finishing a give-and-go with Hedman.
"When [Kucherov] has the puck and he's taking those first three steps, creating space and sucking guys in, that's when he's at his best," Stamkos said. "… When the puck is on his stick it's a very good thing for us."
Stamkos put the Lightning ahead 5-4 at 3:00 of the third period after taking a pass from Ondrej Palat in the slot.
Kucherov made it 6-4 at 6:50 on a one-timer off a Hedman pass, and Mikhail Sergachev pushed the lead to 7-4 at 11:55.
"We had a good third period," Kucherov said. "Just the way we played as a team. If you look at our first shift in the third it was like a different team playing. Hopefully we can build off it."