Lecavalier's goal from the top of the right circle came after the Rangers were penalized for having too many men on the ice. Rangers captain Jaromir Jagr argued with officials to no avail after the penalty was called.
Tampa Bay, which has won 10 of 12, also got goals from Paul Ranger and Brad Richards. Johan Holmqvist made 24 saves.
Jason Krog and Michael Nylander scored for the Rangers, who have lost six of nine. Henrik Lundqvist stopped 38 shots.
It was the Rangers' first game since left winger Brendan Shanahan's comments after a loss to Toronto on Wednesday night in which he called for an end to unfair officiating he feels Jagr endures.
Jagr didn't draw any penalties, which appeared to be correct. He did have an assist.
Tampa Bay, however, got the first four power plays, converting the fourth chance when Richards scored from left circle to give the Lightning a 2-1 lead with 5:54 left in the second. The next two minor penalties were called against Tampa Bay, which resulted in Nylander's tying goal at 18:23 during a 4-on-3 power play.
Referees Don Koharski and Craig Spada ended the second by calling two minors against the Rangers.
The Lightning were 2-of-7 with the man advantage. The Rangers converted one of three power-play opportunities, with the final chance coming with 1:41 left.
Krog scored on the rebound goal after a tough angle shot by Shanahan to put the Rangers ahead 1-0 at 7:31 of the first. Ranger tied it at 1 on a rebound backhander from the high slot at 6:51 of the second.
Notes: Lecavalier has scored a goal in three straight games. ... Rangers D Michal Rozsival (knee) missed his second straight game. ... Tampa Bay captain Tim Taylor (flu) didn't play for the third consecutive game.