The unidentified man was listed in serious condition at a local hospital, Atlanta Fire Rescue Department spokeswoman Janet Ward said.
Details on exactly how the man fell - and what part of the stadium the incident occurred in - weren't immediately available.
Heavy rains had led to a nearly two-hour delay of the game, which was scheduled to start at 7:10 p.m.
A Braves spokeswoman referred calls to the Atlanta police. A police spokesman did not immediately return a message.
Monday's accident wasn't the first of its kind to happen at Turner Field, and marked at least the third time a sports fan has fallen from the stands in Atlanta in about a year.
Isaac Grubb, 20, of Lenoir City, Tenn. died after falling over a railing at the Georgia Dome during a football game between Tennessee and North Carolina State on Aug. 31, 2012. Authorities said he landed on another man seated in the lower level, and that alcohol was a factor.
A man fell about 25 feet over a staircase railing at a Georgia Tech-Miami football game on Sept. 22, 2012 and was not seriously injured.
In May 2008, a 25-year-old Cumming, Ga. man suffered head injuries when he fell down a stairwell at Turner Field during a game against the New York Mets and later died. Police found that alcohol had factored into that accident, which the Braves said was the first non-medical fatality to happen at the ballpark.
Turner Field became the home of the Braves in 1997, a year after serving as the site of major events for the 1996 Summer Olympics.
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