Weaver, playing in his 123rd career game, scored during the Kings' two-goal third.
"It's about time," said Weaver, originally credited with two goals before a postgame scoring change. "I wish we would have had a victory there at the end, but we didn't. I'm just going to enjoy [the goal] for a couple of hours, then forget about it and concentrate on the next game."
Lecavalier and Blair Jones scored first-period goals for the Lightning, who have won 11 of 13. Tampa Bay took the shootout 3-1.
"We weren't the best we could be," Lightning coach John Tortorella said. "We didn't play a very good game, but you still get two points. We're not the only team that goes through that. The teams that win the majority of those are the teams that are there in April."
Tampa Bay took a 2-0 lead in the first period, but the Kings rallied with two goals in the third. Weaver's second was awarded to Derek Armstrong about 40 minutes after the game.
"I was just like, 'Get the puck. Make sure you get the puck from the ref,'" Weaver said. "Points are not my thing anyway, so when I do get them, it's just a bonus."
The Kings, who have the worst record in the Western Conference, are 2-8-3 over their last 13 games.
Weaver cut Los Angeles' deficit to 2-1 at 1:13 of the third. Armstrong tied it on a deflection of Weaver's shot from the blue line with 12:19 left in the period.
Lecavalier put the Lightning ahead 1-0 during a power play at 2:23 of the first. Jones made it 2-0 on his first NHL goal at 4:18.
Tampa Bay's Johan Holmqvist made 29 saves. He has allowed just one goal in 17 shootout attempts this season.
"It looked like we had control for two periods, but they got back in the game," Holmqvist said. "It was a good feeling to win a shootout again."
Tampa Bay is 9-2 in games decided in overtime or a shootout this season.
Los Angeles goalie Sean Burke, who was playing for the Lightning's AHL team in Springfield before being claimed on waivers Jan. 18 by the Kings, stopped 27 shots.
"We had a chance to win that game," said Burke, who is 2-2-2 with the Kings. "I just got beat in the shootout. We got down, but I think we showed a lot of character -- 2-0 in the other team's building in the first seven or eight minutes can be pretty intimidating."
Kings center Michael Cammalleri missed the game with what team officials called a muscle strain. No other details regarding the injury that sidelined the Kings' second-leading scorer (22 goals and 51 assists in 55 games) were provided.
Jones has appeared in 10 NHL games, all this season. ... Lightning left winger Ruslan Fedotenko (flu) didn't play. ... The Kings have been outscored 65-45 in the first period this season. ... Tampa Bay captain Tim Taylor returned after missing the past three games with the flu.