Lecavalier's game-winner came on an easy one-timer next to the net set up by Martin St. Louis' pass across the crease.
The game's other eight goals came during a thrill-a-minute second period, including Steven Stamkos' fourth career hat trick for Tampa Bay.
Brett Connolly also scored for the Lightning, who have won five of their last seven, while Dwayne Roloson made 28 saves. Roloson gave up four goals on nine shots in the second period.
Patrick Sharp scored twice in the second period for Chicago and Marian Hossa and Daniel Carcillo added goals. The Blackhawks, who had a three-game winning streak snapped, played their third straight overtime game.
Ray Emery allowed five goals on 32 shots, including four on 14 shots in the second period.