Kucherov gave Tampa Bay a 1-0 lead at 1:40 of the first period when he redirected a pass from Erik Cernak at the right side of the net.
Paul extended the lead to 2-0 with a power-play goal at 9:05, redirecting a touch pass off a fake shot from Kucherov.
"When you're the ice with him you expect the unexpected," Paul said. "He didn't look at me once and it's right on my tape. Couple of shifts I had before, I kind of learned to keep my stick on the ice. You get in the shot lane, and the next thing you know my stick is open and he slides it right across without even looking. He's a special player."
Gavin Bayreuther cut it 2-1 at 7:04 of the second period with a shot that deflected off the stick of Tampa Bay forward Brandon Hagel.
"First and foremost their power play is lethal," Bayreuther said of the Lightning, who were 3-for-4 with the man-advantage. "Kucherov is one of the best half-wall guys in the League. That being said, I think we gave up again. Guys get frustrated. It's hard, it's really hard to win in this League, but it's really unfortunate."
Vladislav Namestnikov gave Tampa Bay a 3-1 lead on a breakaway at 10:25.
"Goals haven't come easy this year, so you have to take the opportunity and I got one there on the breakaway," Namestnikov said. "I saw I had a clear lane and he was out of position, so I took it to the backhand."