Selected in the seventh round by the Kansas City Royals in 2009, Baumann has spent his entire seven-year career in the minors.
The 28-year-old made 34 appearances (six starts) with Triple-A Omaha last season, going 3-4 with a 3.39 ERA and 1.17 WHIP across 77 innings, striking out 84.
In order to make room on the 40-man roster, the Padres designated catcher Josmil Pinto for assignment. The 26-year-old became expendable following the acquisition of Christian Bethancourt from the Atlanta Braves earlier this month.
Pinto, who was selected off waives from the Minnesota Twins in November, slashed .219/.315/.391 with seven home runs and 50 strikeouts in 57 games last season.