Bryan Rust scored either side of Jonathan Drouin's goal for the Lightning to seal a 2-1 win and the Eastern Conference series.
It is the first time the Penguins have made the Stanley Cup Finals since 2009, when they defeated the Detroit Red Wings in a rematch of the 2008 series.
After a scoreless first period, Rust gave the Penguins the lead one minute, 55 seconds into the second after the Lightning defence fell apart.
The 24-year-old received a pass from Chris Kunitz in the middle of the ice, smashing the puck into the corner of the net unopposed.
Drouin managed to tie the game at 1-1 at 9:36 of the second period, but their jubilation was short-lived.
Just 30 seconds after Drouin scored, Rust struck again for the Penguins to restore their lead, and from that point on they held on.
The Lightning fought hard for a late goal to tie the game at 2-2, but the Penguins defence held firm to seal their spot against the Sharks.