TAMPA, Fla. -- Tampa Bay capitalized on a rare third-period letdown by Calgary.
"It's a huge sign being able to win a game like this," said Tampa Bay right wing Martin St. Louis.
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 made 25 saves and 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.
"They got some momentum and they're a good offensive team," said Calgary captain Jarome Iginla. "Once they get a little jump, especially at home, they start bringing it."
The Flames were coming off a 2-1 win over Florida on Friday night, a game in which they allowed just 13 shots. Miikka Kiprusoff stopped 21 shots Saturday, including all 10 he faced through two periods.
"We'll take the positives and move on," said Iginla.
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 backhand 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 game misconduct stemmed from the right wing's sweater not being tied down.
The teams combined for 14 penalties and 16 shots during the first.
"That was one of the toughest games for grinding it out," said Lightning coach Rick Tocchet. "Everybody contributed. That's the kind of team we have to be."
The Flames are 8-2-2 -- including 3-0-1 in games with Southeast Division teams -- against the Eastern Conference this season.
"We got a point and move on," said Flames coach Brent Sutter. "It was there for us to have two, no question. We came up a little short."
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.
Flames RW Rene Bourque left in the first with an undisclosed injury. ... 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.