He played in Major League Baseball (MLB) for the Atlanta Braves, Kansas City Royals, Texas Rangers, Oakland Athletics, and Cincinnati Reds.

Minor played college baseball at Vanderbilt University and was selected by the Braves with the seventh overall pick in the 2009 MLB draft.

[1] Minor played at Vanderbilt University with fellow future first round draft picks Pedro Alvarez and David Price.

One highlight of the 2008 season was his earning the 4–1 win over Cuba in the championship game of the Haarlem Baseball Week tournament in The Netherlands.

In the 2008 World University Baseball Championship held in Brno, Czech Republic, Minor led Team USA to its third gold medal alongside Stephen Strasburg.

[6] On May 25, 2013, Minor hit his first career home run off Dillon Gee of the New York Mets in a 6–0 Braves win.

[7] On August 22, 2014, Minor pitched 7+2⁄3 no-hit innings, before giving up an RBI single to the Reds' Billy Hamilton.

[11] On March 3, 2015, Minor again began feeling tightness in his shoulder[12] and was diagnosed with rotator cuff inflammation.

[14] Minor attempted to start throwing programs in mid-March[15] and early April,[16] but felt discomfort both times.

[17] Eleven days later, Minor underwent surgery for a torn labrum, and missed the rest of the season.

[29] On August 31, 2020, Minor was traded to the Oakland Athletics in exchange for Dustin Harris and Marcus Smith.