The Braves gave up two minor-leaguers -- left-hander Joey Wentz and infielderTravis Demeritte-- for Greene. They sent right-handed pitchers Dan Winkler and Tristan Beck to the Giants.
The Braves also acquired catcher John Ryan Murphy from the Arizona Diamondbacks on Wednesday for cash considerations.
"We engaged everything -- position players, starting pitchers, the bullpen -- right up until the end,'' Braves general manager Alex Anthopoulos said. "At the end of the day, where we thought there were deals that made sense for us and what we had to give up and so on, the bullpen made the most sense. But we definitely tried some other areas. We just couldn't wind up with a deal that made sense to our organization.''
Greene, 30, has a 1.18 ERA and 22 saves this season. The right-hander recorded 32 saves last season in his first full year as a closer.
Melancon, 34, is a three-time All-Star who has a 3.50 ERA in 43 games with the Giants this season.
Greene and Melancon are both under contract through the 2020 season. Greene is eligible for arbitration, and Melanconis set to make $14 million in the final year of his big deal, which the Braves will fully assume.
"That was part of it, no doubt,'' Anthopoulos said. "That adds value. You want contractual control."
Melancon waived a no-trade clause to join the Braves after his wife gave the green light.
"She's running the show here,'' Melancon said after his bags were packed in the visiting clubhouse at Philadelphia. "After we talked and realized it's Atlanta, a really good situation over there, we decided it's OK. It's a winning team. They got a lot of young talent, a lot of upside there."
The Braves bolstered their bullpen Tuesday with their acquisition of right-hander Chris Martin from the Texas Rangers.
Anthopoulos said the flurry of moves at the trade deadline pushed Atlanta well over its planned budget for 2019, but the GM was quick to point out that the much-maligned ownership group signed off on the extra spending without much hesitation.
Liberty Media has come under fire in the community for zealously guarding its checkbook, despite getting a new suburban ballpark that was largely paid for with some $400 million in public funding.
"We're a significant chunk over our allocated budget for the current year,'' Anthopoulos said.
Luke Jackson has served as Atlanta's closer this season, recording 17 saves. However, he has struggled lately, posting a 13.50 ERA with two blown saves in seven appearances since the All-Star break. He blew a save chance on Wednesday against the Washington Nationals, but the Braves won 5-4 in 10 innings.
For the Tigers, the trade of Greene likely clears the closer role for Joe Jimenez, who has been the team's closer of the future for some time. Buck Farmer, who has nine consecutive scoreless appearances to begin the second half, could force himself into the mix if Jimenez initially struggles.
The Associated Press contributed to this report.
Teixeira: Braves needed a guy like Greene
Mark Teixeira and Tim Kurkjian both like the Braves' addition of Shane Greene to solidify their bullpen.