Tkachuk, who returned from a lower-body injury, scored twice in the second period, the first half of a quick 14-second double punch that gave Florida a 4-1 lead.
Jake Guentzel and Brayden Point scored for the Lightning, who lost center Anthony Cirelli halfway through the second period. There was no word on his status. Andrei Vasilevskiy stopped 10 shots in goal for Tampa Bay.
Bennett got on the board early in the first by batting a loose puck home. Guentzel tied it up on the power play, but a tip-in by Reinhart had Florida in front 2-1 before the period was through.
Guentzel made waves online for his antics following his power-play goal, where he really gave it to Tkachuk as he exited the penalty box and made his way to his team's bench.
Florida opened it up in the second. After Schmidt's rebound goal at 4:41, Tampa Bay's failed goaltender interference challenge gave the Panthers a power play, and Tkachuk capitalized just seconds later.
He struck again at 9:44, before Point briefly cut the deficit to 5-2. Schmidt sealed it with Florida's third power-play goal in the third, thanks to an assist from Tkachuk. Game 2 is scheduled for Thursday night at Amalie Arena.