Justin Hayes of Cumming, Ga., suffered head injuries and was taken to Grady Memorial Hospital.
The investigation is "pointing toward drinking. Alcohol was a factor," Atlanta police spokesman Ronald Campbell said Thursday.
Hayes' parents said Justin Hayes had "a couple of beers."
"According to his friends, he had a couple of beers like most people at a game," said John and Cindy Hayes in a statement released Thursday. "They were leaving and having fun, and he did what he does at home -- tried to slide down the railing. Unfortunately, he just slipped over."
Campbell said Hayes fell about 150 feet from the club level to the landing on the field level during the eighth inning. Braves spokesman Brad Hainje said Thursday the fall was about 50 to 60 feet.
"This was a tragic, horrible accident," Hayes' parents said. "Justin was a very strong Christian person, from a strong Christian family, who had never been in any trouble. He really cared about people and went out of his way to help anyone who needed it. We realize most parents will say that about their child, but that was honestly how he was."
Mark Guilbeau, the senior investigator with the Fulton County Medical Examiner's Office, said an autopsy will be performed.
The Braves issued a statement Thursday expressing sympathy for Hayes' family.
"The Atlanta Braves and Atlanta Police Department are investigating the tragic accident that resulted in the death of a 25-year old male fan at last night's game," the statement said. "Our sincerest and heartfelt condolences go out to his family."
Braves spokeswoman Beth Marshall said the team had no additional comment pending the investigation.
Turner Field, built for the 1996 Olympics, has been the Braves' home stadium since 1997. The Braves said it's the first non-medical fatality at the stadium.