The Braves announced the moves Wednesday. Atlanta gave up right-handed pitchers Dan Winkler and Tristan Beck for Melancon. The Braves sent minor-leaguers Joey Wentz and Travis Demeritte to the Tigers for Greene.
Greene, who was named to his first All-Star Game in 2019, has 22 saves and a 1.18 ERA with the Tigers this season. The 30-year-old reliever recorded 32 saves in 2018, his first full season as a closer.
Melancon, 34, is a three-time All-Star selection. He has a 3.50 ERA in 43 games with the Giants this season.
Melancon and Green both remain under contract through the 2020 season. Melancon has a base salary of $14 million, and Greene is eligible for arbitration.