Johnson, 31, went 5-2 with a 7.09 ERA in a combined 53 1-3 innings last year for Oakland and Detroit. The former Baltimore closer is 23-28 with a 3.57 ERA and 124 saves in his career.
The Braves hope Johnson can replace Jordan Walden as the primary setup man for closer Craig Kimbrel. Walden was traded with right fielder Jason Heyward to St. Louis last month in the deal that brought right-hander Shelby Miller to Atlanta.
Johnson, an AL All-Star two years ago, led the majors with consecutive 50-save seasons in 2012-2013. He was originally drafted by the Orioles in the fifth round of 2001.