Hahn was called up from Double-A San Antonio, where he compiled a 2.20 ERA with a 2-1 record across 10 games (seven starts) with the Missions.
The 24-year-old right-hander was acquired by the Padres in January along with pitcher Alex Torres in a seven-player trade with Tampa Bay.
San Diego also placed pitcher Eric Stults on the bereavement list, selected the contract of pitcher Jason Lane from Triple-A El Paso and optioned pitcher Donn Roach to the Chihuahuas.
Stults, who lost his last start against the White Sox on Sunday, is 2-6 with a 5.03 ERA in 12 starts for the Padres. He can stay between 3-to-7 days on the bereavement list.
Lane, who last played in the majors as an outfielder with the Padres in 2007, has recorded a 4.36 ERA with a 5-4 record in 11 starts at El Paso. Roach is 1-0 with a 4.75 ERA in 16 appearances with San Diego this season.