"In Drew Stubbs, we received a great defensive player who brings an element of speed and power to our lineup," Antonetti said. "We are happy with the job Michael Brantley did for us in center field, but Drew is one of the best out there. He brings a great deal of athleticism to our team."
Bauer, one of baseball's top pitching prospects, became the first member of the 2011 draft class to reach the majors and went 1-2 with a 6.06 ERA in four starts for Arizona this year. The 21-year-old right-hander struggled with his control, compiling 13 walks and 17 strikeouts in 16 1-3 innings.
Bauer spent most of the season in the minors, going a combined 12-2 with a 2.42 ERA in 22 starts at Double-A Mobile and Triple-A Reno. He had 157 strikeouts and 61 walks while allowing 107 hits in 130 1-3 innings.
"We are getting a young pitcher with a ton of potential," Antonetti said. "He can be a top-of-the-rotation guy with development. He is somebody we have been interested in for a long time, since he was at UCLA."
Arizona general manager Kevin Towers said he was sorry to part with Bauer, but "we're fortunate to have a lot of pitching depth."
The 22-year-old Gregorius, considered a defensive whiz, spent last season with Double-A Pensacola and Triple-A Louisville before appearing in eight games for the Reds. He hit a combined .265 with seven homers and 54 RBIs in 129 minor league games, adding 21 doubles and 11 triples while scoring 70 runs.
The Diamondbacks have been looking for a shortstop to replace Stephen Drew, traded to Oakland in August after returning from a serious ankle injury. They were thought to be interested in Indians All-Star Asdrubal Cabrera, though Towers wouldn't comment on that.
Towers said Gregorius reminds him "of a young Derek Jeter." Gregorius will go to spring camp and compete for the starting job in the big leagues.
Arizona completed a $15.5 million, two-year contract with right-hander Brandon McCarthy on Tuesday and also acquired reliever Heath Bell from Miami earlier this offseason. Towers said those additions make it "highly unlikely" the team will trade slugger Justin Upton.
AP Sports Writer Mike Fitzpatrick and AP Baseball Writer Ben Walker in New York contributed to this report.