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.
The Padres have plenty of outfielders, including Carlos Quentin, Cameron Maybin, Will Venable, Chris Denorfia and Alexi Amarista, who had been primarily an infielder.
Quentin and Maybin missed significant time with injuries last season.
Padres general manager Josh Byrnes said Smith could play left or right field, and probably will end up with 400 to 500 plate appearances. Smith will allow the Padres to match up better against right-handers and have a more viable bat in the lineup when Quentin needs days off due to his balky knees.
"Obviously he did a great job this past year and hit lefties better than he ever has, which is a good thing," Byrnes said of Smith. "We liked our offence for a year, kind of from All-Star break to All-Star break. Injuries last year, toward the end of the year we weren't scoring runs. The reality is, we're coming off a year where we asked Venable and Denorfia to do more than they ever have. We asked Amarista to play a lot in the outfield. We extended Quentin and Maybin.
"I don't think we've overdone it by acquiring Smith, but we just want to make sure that if there is an injury or two, that we don't have such a dropoff, that we're still showing up every night, especially against right-handed pitching, and able to score runs," Byrnes added.
Byrnes said the Padres might be looking to add a reliever or two, or could move some starters to the bullpen.