Malkin tied it 2-2 at 6:17 of the second period, receiving a stretch pass from Josh Archibald as he exited the penalty box before scoring five-hole on a breakaway.
Crosby put the Penguins in front 3-2 with a power-play goal at 15:28, finishing a backhand pass from Jake Guentzel at the right post.
"Obviously, take that penalty there and they score on the power play," Lightning captain Steven Stamkos said. "Kind of sucked the life out of us a little bit. I don't know the details, but that was obviously tough. I thought it was a decent game before that."
Teddy Blueger made it 4-2 at 16:20 when he dove to knock in a loose puck after Marcus Pettersson's shot caromed back into the crease off the end boards. It was just his second goal this season and first since Dec. 6 (33 games).
"Just overall, good team effort," Blueger said. "Had to fight through it. It's obviously a good feeling to score at any time, but especially when you haven't done it in a while."
Jeff Carter extended the lead to 5-2 at 17:50 with a wrist shot from the right circle, and Jason Zucker scored 47 seconds later on a backhand in front to make it 6-2.
Dumoulin pushed it to 7-2 at 19:55 with his first goal of the season, a slap shot from the point off a face-off win by Crosby.
"Didn't give our goalie a chance to save some of those pucks," Tampa Bay defenseman Mikhail Sergachev said. "That's what happened."