The Braves also received cash in the deal.
Swisher, 34, who is on the 15-day disabled list because of inflammation in his left knee, has struggled over the past two seasons batting just .208 with eight home runs and 42 RBIs in 97 at-bats in 2014, and .198 with two home runs and eight RBIs in 30 at-bats this season.
Bourn, 32, is batting .246 with 12 doubles and 13 stolen bases in 2015. He led the AL in triples with 10 last year.
Johnson, 30, is batting .235 with two home runs and 11 RBIs in 56 games this season.
The Braves and Indians had previously discussed a potential trade involving Swisher and Johnson in July, but Atlanta backed out because of Swisher's salary.
Bourn and Swisher are due a combined $29 million in 2016, while Johnson is due $16.5 million through 2017.
- Kayla Lombardo