Glaesmann, 23, hit .240 with 11 home runs and 54 RBIs in 132 games for Double-A Montgomery in 2013.
Glaesmann completes the Dec. 3 trade that also saw the Rays send Minor League right-hander Justin Choate to the D-backs. The Reds received left-hander David Holmberg from the D-backs in the deal.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.