Jonathan Quick made 35 saves for the Kings (18-25-5), who have lost five of their past six.
Jeff Carter gave the Kings a 1-0 lead at 4:18 of the first period on a tip-in, and Kyle Clifford made it 2-0 at 6:52.
"Overall great performance by our team," Kings defenseman Sean Walker said. "It's tough blowing a lead there a couple of times, but I think overall you look at the game and I think we played great. There's lot of opportunities we probably could've scored there again, but give credit to them, they've got a great goalie."
Killorn made it 2-1 at 10:06 with his 19th goal of the season to match his NHL career high. Stamkos tied it 2-2 with a goal off a rebound at 11:25.
Dustin Brown gave the Kings a 3-2 lead with a power-play goal at 12:00 of the third period.
"[Losing] shootouts ... it is what it is. I think we played a really good game, but it just happens," Brown said. "They find a face-off goal, it's not too often a guy wins it and it's a one-timer from the middle of the ice, but they have the ability and put it in our net."
Cedric Paquette appeared to score a game-tying goal with 6:05 remaining in the third period, but it was waved off when the officials determined the Lightning forward hit the puck with a high stick. Video review supported the call.
"As a team they don't give you much, so it's a bit of a waiting game," Lightning defenseman Kevin Shattenkirk said. "[Quick] is a goalie that can make it tough on you to score goals, we just had to stick with it until the end and we finally got one to go through on the 6-on-5. It was a great shot by [Kucherov] to put that in."
Vasilevskiy made a save against Brown in the shootout to finish the win.
"I scored on that move plenty of times in my career, but it's generally against shorter goalies," Brown said. "Thinking about it now it's a harder move to pull off on a longer-legged goalie, which he is (6-foot-3)."
Carter's goal ended Vasilevskiy's shutout streak at 150:13. He has not lost since Dec. 14 and is 13-1-1 since Dec. 7. He was named to the NHL All-Star Game on Monday, replacing Boston Bruins goalie Tuukka Rask.
"He's the best goalie in the League," Kucherov said. "He should've gotten picked in the first place."