Brandon Hagel set up Kucherov for a tap-in on a 2-on-1.
"Some big-time players made some big-time plays down the stretch for us," Lightning coach Jon Cooper said.
"I think we could even be better, but we were progressing the right way," Killorn said.
"I feel a lot better about our team and the direction we're going in than I did a week ago," Sharks coach David Quinn said.
A costly mistake leading to the odd-man rush for the winner was something of focus inside the Sharks' locker room.
"In order to win games in this league, you're going to have to take some risks," Karlsson said.
Hedman opened the scoring 32 seconds into the game, and Karlsson tied it 50 seconds later on a blast from the point.
Colton gave Tampa Bay a 2-1 lead with 6:16 left in the first period. Sturm tied it with his fifth of the season at 5:36 of the second, and Killorn put the Lightning back in front with 1:19 left in the second.
"It's been nice just to get some chemistry going," Colton said.
Killorn scored with under two minutes to go in the second to give the Lightning a 3-2 lead. It was Killorn's second goal of the season.
Karlsson tied it again on a power play at 4:34 of the third. He has six goals to lead the Sharks.
"(Karlsson's) a very special player. He's very important to our team. He's a leader, especially in the way he plays and that's what we need. We expect nothing less," Mario Ferraro said.
Quinn remains a victory shy of the century mark.
Sharks: Continue a six-game homestand against Anaheim on Tuesday night.
Lightning: Begin a four-game homestand against Ottawa on Tuesday night.
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