Stanton was drafted by the Atlanta Braves and he made his major league debut in 1989 with the Braves, where he spent the first six years of his career. He pitched in 20 games his first year, racking up 27 strikeouts and seven saves. Stanton struggled mightily in his sophomore season, going 0–3 with an ERA of 18.00 in just seven innings (giving up 14 runs). 1991 was his first year as a full-time player, and he put up a 5–5 record in 74 games, earning a spot on the postseason roster. He pitched well, allowing just one run in eight total games, including a victory in the 1991 World Series. In 1992, he went 5–4, (his first season with a record over .500). The Braves once again lost the World Series, this time to the Toronto Blue Jays, as Stanton allowed three hits and two walks during the series. The next year, he fashioned a 4–6 record with a in 63 games, but earned a career-high 27 saves. He pitched one inning of one-hit, one-walk baseball in the 1993 National League Championship Series. In 1994, Stanton fashioned a 3–1 record before the strike ended the season early. He was granted free agency following the season, but re-signed on April 12, 1995. When play resumed in 1995, Stanton pitched in 26 games, garnering a 1–1 record and 5.59 ERA before being traded on July 31 to the Boston Red Sox.