Kane scored again at 15:54 to make it 2-0. With the Blackhawks on the power play, he took a one-timer at the goal line off a feed from Cody Franson.
"We had a great first period, played with great pace," Blackhawks coach Joel Quenneville said. "It was an exciting game, we had our chances to get the lead again. They're a dangerous team and we took one too many penalties."
Palat made it 2-1 at 4:27 of the second period with a shorthanded goal. Victor Hedman forced a turnover at the blue line and dropped a pass for Callahan, who set up Palat for a one-timer.
Kunitz tied it 2-2 31 seconds into the third period when he went to the five-hole to beat Crawford.
"That line had energy tonight, I thought that Cedric Paquette coming back in the lineup helped tremendously," Lightning coach Jon Cooper said. "Obviously we don't come back and win that game without that line."