Lecavalier has a career-high 39 goals, three shy of Brian Bradley's team record set during the 1992-93 season. Lecavalier has scored in 15 of 19 games.
Washington's Dainius Zubrus forced overtime by scoring with 14.9 seconds left in the third.
The Lightning, who appeared earlier this season in jeopardy of missing the playoffs just three years after winning the Stanley Cup, have won 14 of 17 and are one point behind the Southeast Division-leading Atlanta Thrashers.
Alexander Semin also scored for the Capitals, who have lost 11 of their last 12 road games. Washington played for the first time without injured goalie Olie Kolzig, who hurt his knee in practice Monday and is expected to miss at least three weeks.
Semin scored his 30th goal during a power play to put Washington ahead 1-0 at 9:07 of the first.
Lecavalier tied it 1-1 with a power-play goal at 2:44 of the second. He then made it 2-1 on his career-best third short-handed goal of the season with 12:55 left in the period.
Washington all-star Alexander Ovechkin didn't record a point for the fourth time in six games. Tampa Bay goalie Marc Denis made a glove save on Ovechkin's shot from the left circle during a power play midway through the third.
Ovechkin has one goal and an assist over the past six games. He was one of three Capitals who failed to score on Denis during the shootout.