Swarzak, 33, had a 5.27 ERA in 15 games for the Mariners this year. He struck out 17 hitters in 13 2/3 innings.
Swarzak has a 4.33 career ERA in 10 seasons. He had a 6.15 ERA in 29 games with the Mets last year.
The Braves traded injured closer Arodys Vizcaíno and left-hander Jesse Biddle to Seattle. Vizcaíno is out for the rest of this season after undergoing right shoulder surgery in mid-April.
Biddle had a 5.40 ERA in 15 games for the Braves this season. The team designated him for assignment last week.
Atlanta is 25-22 this season. The Braves are 2 1/2 games behind Philadelphia in the NL East Division.