The goal was Meszaros' fifth this season.
His blistering wrist shot nearly two minutes into the extra session ended the game.
The contest was tied, 1-1, through three periods, with Tampa Bay's Steven Stamkos (32) scoring in the first minute of the second after Calgary's Mark Giordano (10) (power play) had scored at 15:44 of the first.
Goalie Antero Niittymaki made 25 saves on 26 shots for the Lightning, who have won three in a row and six of eight.
Net-minder Miikka Kiprusoff allowed both goals on 23 shots for the Flames, whose modest two-game winning streak ended.