Thompson put the Sabres ahead 2-1 with 10 seconds left, sliding in a shot into the far side of the net off a pass from Zemgus Girgensons on a 2-on-1.
Buffalo outshot Tampa Bay 16-6 in the period.
"If you had watched us play Carolina (in a 3-1 home win on Thursday), it was the exact same thing," Lightning coach Jon Cooper said. "We sleepwalked through the first period, especially [against] a team that had to travel and come in on a back-to-back and we just didn't execute. We thought it was the first day of training camp. But to be honest, I left them alone and let the leaders take over in the second period."
Palat tied it 2-2 at 7:50 of the second period after Mikhail Sergachev faked a shot before passing to him.
"Anytime he can get the puck at the blue line and has that shot fake that buys him some extra time, that's a huge part of his game," Stamkos said of Sergachev, who is in his third NHL season. "That's what we need from him on the offensive side. He's shored up a lot defensively in the short time he's been in this league already and he's going to continue to get better."