The deal announced Sunday includes a prorated salary. Greene missed the first 38 days of the season, so he will receive about $1.19 million.
Greene agreed to be optioned to Triple-A Gwinnett, and he will need time to get ready before returns to the majors. The right-hander went 1-0 with a 2.60 ERA and 21 strikeouts in 27⅔ innings last season.
Braves relievers began Sunday with a 4.56 ERA that ranked 22nd in the majors. It's a considerable slip from last year, when the bullpen ranked fourth with a 3.50 ERA.
"He showed the benefit in the role he assumed in that very strong bullpen last year," Braves manager Brian Snitker said of Greene before the game, when the deal was not yet official. "There's a guy that's willing and able to do pretty much any role in that bullpen. He came to me last year when we had all the starting woes and guys were down and said, 'I'll start games if you want me to.'
"He's a durable guy. He had a very solid year. If and when we get him back here, I'll be excited to get him in the mix."
The Braves are expected to get reliever Chris Martin back on Tuesday. He has missed the past 29 games with right shoulder inflammation.
"These guys have done a great job this last week," Snitker said of his relievers. "They've been very big in that series in Washington. And last night, what those guys did, they've done a really strong job here, handing the ball off to each other and what they've done. But if you add guys like that, absolutely, it's going to help strengthen the bullpen."
Atlanta originally acquired Greene in a three-player trade with the Detroit Tigers on July 31, 2019, and he has pitched 55 games for the Braves over the past two seasons, going 1-1 with a 3.27 ERA and one save.
Greene allowed seven runs over his first six appearances with Atlanta after being acquired in 2019. But in his past 49 games for the club dating to Aug. 14 of that season, he has allowed just 12 runs over 48.0 innings for a 2.25 ERA and 38 strikeouts.
The Associated Press contributed to this report.