Meszaros' winner came from the left circle to complete Tampa Bay's comeback from a 1-0 deficit after the second period.
Steven Stamkos also scored for the Lightning, who have won seven of 10. Goalie Antero Niittymaki improved to 6-0-1 over his last seven starts.
Calgary got a goal from Mark Giordano. The Flames dropped to 23-2-2 when leading at the start of the third.
Giordano put the Flames up 1-0 on a power-play goal with 4:16 left in the first. Stamkos tied it at 1 with a rebound goal a minute into the third.
Stamkos has seven goals and 14 points during a nine-game point streak.
Calgary's Ales Kotalik received a fighting major and game misconduct after a fight with the Lightning's Stephane Veilleux late in the first. The teams combined for 14 penalties and 16 shots during the first.
Calgary and Tampa Bay have had little success since meeting in the 2004 Stanley Cup finals. Neither has won a playoff series since, and the Lightning finished last and next-to-last overall the past two seasons, respectively.
NOTES: Lightning RW Martin St. Louis had his streak of a goal in five consecutive games end. He did have an assist. ... Flames C Daymond Langkow played in his 1,000th NHL game. ... Calgary resident Lou Jankowski, who played for Detroit and Chicago in the early 1950s, was at the game.