The Padres made room on the 40-man roster for Kelley by designating right-hander Keyvius Sampson for assignment.
Kelley went 3-6 with four saves and a 4.53 ERA in 59 relief appearances in 2014. In five big league seasons with Seattle and New York, he has a 17-17 record with four saves and a 3.94 ERA.
Barbato went 2-2 with 16 saves in 18 opportunities and a 2.87 ERA in 27 games with Double-A San Antonio.
Sampson went 2-5 with a 6.68 ERA in 38 games, including 14 starts, for Triple-A El Paso.