Anthony Cirelli scored for Tampa Bay, which has won its past five season openers.
"I thought we had a really good second period. I really like the way we were playing," Lightning coach Jon Cooper said. "But you just can't take three penalties in the third period. It sucked the life out of us. It's something we're obviously going to have to address now."
Jacob MacDonald scored in his NHL debut for Florida. Roberto Luongo made 13 saves before leaving the game with a lower-body injury at 12:21 of the second period. He was replaced by Reimer, who made 15 saves.
MacDonald, who replaced MacKenzie Weegar (shoulder) in the lineup, beat Vasilevskiy short side from the left circle on his first shot at 5:58 of the first period to give the Panthers a 1-0 lead.
"It's obviously my dream to play and have a shift like that my first time out is awesome," MacDonald said. "It was a lot of fun coming in here and they have a great environment."
Cirelli tied it 1-1 with a shorthanded goal at 10:45 of the third period after McDonagh forced a turnover and passed to him for a 2-on-1. Cirelli tried to set up Alex Killorn, but the pass deflected off Panthers forward Aleksander Barkov's stick back to Cirelli, who scored into an open net.