"It was nice we were able to get that," Johnson said.
Tampa Bay, 3-5-1 in the last nine games, also got a goal and two assists from Nikita Kucherov. Ondrej Palat assisted on all three Johnson goals in the Lightning's first win over Pittsburgh since Nov. 17, 2011.
"It was pretty impressive," Tampa Bay coach Jon Cooper said.
Johnson scored twice in the first and extended the Lightning's lead to 4-0 at 2:37 of the second with a 2-on-1 goal against Jeff Zatkoff, who replaced Marc-Andre Fleury earlier in the period.
Patric Hornqvist, Chris Kunitz and Kris Letang scored for the Eastern Conference-leading Penguins, who have 13 players out due to illness or injury. It was Pittsburgh's (16-1-5) first regulation loss to an Atlantic Division team since getting beat 5-1 Jan. 20 by Florida.
"You don't like to lose, but we definitely had a lot of things going against us," said Crosby, who went without a point. "It would have made it easy to get ready for Christmas break early. I think guys competed hard. Everybody tonight stuck with it, gave ourselves a chance to get back in the game. Made some mistakes, but we didn't give up and gave ourselves a chance."
Pittsburgh forwards Steve Downie and Brandon Sutter, and goalie Thomas Greiss were sent back to Pittsburgh Monday for mumps testing. Penguins coach Mike Johnston said final test results should be available in the next couple days.
Johnson opened the scoring 6:54 into the game from the slot and made it 2-0 on a nifty shot from the top of the left circle with 3.4 seconds remaining in the first.
Andrei Vasilevskiy made nice saves on first-period shots by Malkin and Crosby, who also had the goalie stop his shot on an early third-period breakaway. The rookie goaltender has won two of three starts since being called up Dec. 16 from Syracuse of the AHL following an injury to Tampa Bay starting goalie Ben Bishop.
"I got some confidence," Vasilevskiy said. "Keep working hard and do my best."
Kucherov put the Lightning up 3-0 and ended Fleury's night on a breakaway backhander 52 seconds into the second.
Hornqvist scored at 10:15 of the second. Kunitz had a breakaway goal midway through the third before Letang cut the deficit to 4-3 with 1:31 to play.
"It was only about a 30-minute effort," Letang said. "You get in a hole early when you're not competing well. They took it to us with a couple of really nice shots. Hats off to them for executing and really bearing down on their chances. We tried to make it close but didn't pull through in the end."
Malkin had two assists, giving him 20 assists and 38 points in 29 games against Tampa Bay.
NOTES: Johnston said F Blake Comeau, hurt in Monday night's 4-3 shootout loss at Florida, will miss several weeks with an upper body injury. ... Bishop and D Jason Garrison, both out with lower body injuries, could return right after the Christmas break. Bishop had a 30-minute on-ice workout after Tuesday's morning skate. ... The Penguins recalled F Dominik Uher from Wilkes-Barre/Scranton of the AHL. Uher became the sixth Penguins player to make his NHL debut this month, joining Scott Wilson, Bryan Rust, Bobby Farnham, Scott Harrington and Derrick Pouliot.