The Lightning led 4-0 midway through the game, then held on after the Penguins scored three unanswered goals.
"That line hasn't really been together for a couple of weeks now," Ligthtning coach Jon Cooper said of Palat, Johnson and Kucherov. "But they are finally back healthy. They showed what they can do, and it was really impressive."
Pittsburgh (22-7-5) has lost two straight (0-1-1) entering the three-day NHL holiday break and is 6-3-2 in December. Patric Hornqvist, Chris Kunitz and Kris Letang scored for the Penguins, who have 10 players on injured reserve.
"We definitely need this break to get some guys back here after Christmas," Penguins center Sidney Crosby said. "It's been a tough run here in the past month or so, and it feels like every game we are losing somebody. So hopefully we can get healthy and keep competing the same way."
Johnson completed his hat trick at 2:37 of the second period when he took a pass from Kucherov inside the faceoff circle and sent a wrist shot past Penguins goalie Jeff Zatkoff, who replaced starting goalie Marc-Andre Fleury less than a minute into the second after Kucherov's goal.
"I like to think I can shoot the puck if I need to," Johnson said. "Tonight I had the opportunity and [Kucherov] made some nice plays to give me space to make things happen. I just shot when I saw an opening because the goal was to get the puck on net."
Fleury made seven saves on 10 shots.
Johnson and Kucherov also connected at 6:54 of the first period to give the Lightning a 1-0 lead. Kucherov hit Johnson with a centering pass from the slot, and Johnson sent a one-timer past Fleury on the short side.
Johnson scored his second goal with four seconds left in the first.
"We definitely wanted to get off to a good start against these guys because we've had a tough time with them recently," he said. "And they showed a lot of resiliency by coming back in the third period. We needed everything we were able to get on offense."
Kucherov put the Lightning ahead 3-0 when he scored his 14th goal 52 seconds into the second period after defenseman Matthew Carle's outlet pass sent him in on Fleury on a breakaway.
"We were just trying to go out there and have some fun," Kucherov said. "Tyler has a great shot and he was able to finish everything tonight. It's very enjoyable to have the three of us playing like that and getting those opportunities to help the team win before this break."
The Penguins got on the board at 10:15 of the second period when Hornqvist put a wrist shot over Lightning goalie Andrei Vasilevskiy's shoulder from the left faceoff circle.
"You don't like to lose, but we definitely had a lot of things going against us," Crosby said. "I think the guys competed hard … gave ourselves a chance to get back into the game. We made some mistakes, but we didn't give up."
Following Kunitz's goal at 9:55 of the third period, Pittsburgh pulled to within a goal at 18:29 of the third period when Letang scored on a slap shot from the point after the Penguins had pulled Zatkoff for a 6-on-5 advantage.
"We don't want to find excuses, obviously we have a good side," Letang said. "They had good bounces and scored big goals. … They scored some big goals at key moments. But we hung in there and were one shot away from tying the game."
Vasilevskiy made 26 saves to earn his second NHL win for the Lightning, who defeated the Penguins for the first time since November 17, 2011. Zatkoff made 16 saves.