Fleury's shutout came in his 63rd playoff appearance and got the Penguins off to a solid start in their bid to win the Stanley Cup for the second time in three years. He played in 24 games without a shutout during Pittsburgh's title run in 2009.
Tampa Bay's Dwayne Roloson matched Fleury through two periods but Pittsburgh dominated the final stanza while outshooting the Lighting 12-7.
Alexi Kovalev broke up the scoreless tie 6:05 into the third and Arron Asham scored 18 seconds later on a wraparound shot.
Chris Kunitz put the puck into an empty net with 42 seconds remaining. Fleury made 28 saves overall and Roloson stopped 37 shots.
Pittsburgh will host the second game of the best-of-seven Eastern Conference quarterfinal series Friday.
Notable deaths of 2014 [PHOTOS]