It marked the second swap of the day for the two-time defending AL West champion A's and third in two days.
Gregerson will join a bullpen with new closer Jim Johnson, acquired from Baltimore on Monday night. Dealing Smith became a viable option when general manager Billy Beane acquired Craig Gentry from division rival Texas earlier Tuesday.
Gregerson went 6-8 with a 2.71 ERA and four saves in 73 relief outings for the Padres in 2013.
Smith batted .253 with eight home runs and 40 RBIs in 117 games in his second season with the A's.