Maxime Talbot and Arron Asham also scored for the Penguins, who took a 2-1 lead in their first-round Eastern Conference best-of-seven playoff series and regained home-ice advantage. Game 4 is Wednesday night in Tampa.
Pittsburgh rebounded from a 5-1 home loss in Game 2 despite giving up two more power-play goals to Martin St. Louis. The Lightning star erased a 2-0 deficit by striking late in the opening period, then again early in the third to give Tampa Bay hope of taking control of the series.
But Kennedy answered with the go-ahead goal just 31 seconds later, scoring in heavy traffic from in front of goalie Dwayne Roloson after Pittsburgh won a faceoff in the left circle.
The Penguins have won six consecutive Game 3s and are 10-1 in the third game of playoff series dating to the 2008 post-season. Over the same stretch, Fleury is 12-4 following a playoff loss.
Talbot got the Penguins off to a fast start, scoring just seconds after Tampa Bay's Brian Downie excited a sellout crowd with a hard hit on Ben Lovejoy, levelling the Pittsburgh defenceman behind the Lightning net -- but not before Lovejoy got the puck to Talbot who quickly moved up ice to score at the other end.
Asham made it 2-0 just 45 seconds later, taking a perfect pass from Michael Rupp and tapping the puck into the left side of the net.
After Tampa Bay scored twice on power plays in Game 2, Pittsburgh felt it was essential to stay out of the penalty box to minimize opportunities for the Lightning to make one of the NHL's most effective power plays a factor again.
The Penguins felt they lacked discipline in Game 2, when they thought they allowed Tampa Bay to goad them into several costly penalties and go 2-for-6 on power-play opportunities. The Lightning were 2 of 4 on Monday night, with St. Louis scoring at 15:19 of the first period and again at 2:12 of the third.
Pittsburgh was 0 for 2 on the power play, dropping to 0 for 15 for the series.
Notes: Penguins C Sidney Crosby, out since early January with a concussion, had an off-ice workout, but didn't take part in Thursday morning's optional skate. "His workout regiment, the one he's gone through over the last five days, continues to be all through the doctor," Pittsburgh coach Dan Bylsma said. "There's no change in his status." ... Lightning LW Ryan Malone left the game during the second period with an upper body injury but returned. ... The Lightning's Steven Stamkos recorded the first playoff point of his career with an assist on St. Louis' first goal.