Minor had an outstanding high school career at Forrest School in Chapel Hill, Tennessee and was drafted in the 13th round of the 2006 Major League Baseball Draft by the Tampa Bay Devil Rays after his senior season. However, he did not sign, choosing to attend Vanderbilt University.
Minor played at Vanderbilt University with fellow major leaguers Pedro Alvarez of the Pittsburgh Pirates and David Price of the Boston Red Sox, all of whom were first round draft picks.2007 Collegiate Baseball Freshman All-American
2007 Rivals.com Freshman All-American
2007 SEC All-Freshman Team
2007 SEC Freshman of the Year
2007 Second Team All-SEC
2008 Best pitcher Haarlem Baseball Week
2008 National Collegiate Baseball Writers Association Second Team Pre-season All-America Team
2009 Louisville Slugger Second Team Pre-season All-American
2009 National Collegiate Baseball Writers Association Second Team Pre-season All-America Team
2009 SEC Pitcher of the Week – Week 8
Minor pitched for the United States National Team (Collegiate) in 2007 and 2008.
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. This victory marked the first time in recorded history that a United States Collegiate National Team has defeated the Cuba Olympic Team in a tournament title game. In the 2008 World University Baseball Championship held in Brno, Czech Republic, Minor led Team USA to its third gold medal alongside Stephen Strasburg. In the tournament, he was 1–0 with a 1.15 ERA and 16 strikeouts, allowing 8 hits in 15 2⁄3 innings for Team USA.2008 Baseball America’s Summer Player of the Year
Minor was drafted by the Atlanta Braves in the first round (7th overall) in the 2009 Major League Baseball Draft. and represented the United States in the 2010 All-Star Futures Game.
Minor made his major-league debut on August 9, 2010 against the Houston Astros. He recorded his first strikeout against Chris Johnson. On the night, he went 6 innings giving up 5 hits and 4 runs (3 earned) while walking only 1 and striking out 5 batters; he got no decision. In his next start on August 17, 2010, Minor went 6 innings giving up 5 hits and 2 earned runs and got his first major league victory. Then, in his third career start against the Chicago Cubs, Minor went 6 innings giving up 3 earned runs while striking out 12 and earning his second career victory. Those 12 strikeouts set an Atlanta Braves rookie strikeout record in a single game; surpassing Tommy Hanson's 11 strikeouts in 2009. On August 18, 2011, in a game against the San Francisco Giants, Minor pitched six scoreless innings including facing the minimum number of batters over the last four innings. This victory was the first time in 17 major league starts that Minor did not give up a run. On May 25, 2013, Minor hit his first career home run off Dillon Gee of the New York Mets in a 6–0 win. On August 22, 2014, Minor pitched 7 2⁄3 ho-hit innings, before giving up an RBI single to the Reds' Billy Hamilton. The Braves eventually won the game in the 12th inning on Justin Upton's two-run home run. Minor finished the 2014 season with a 6–12 record and 4.77 ERA, pitching most of the year with a sore shoulder. In 2015, Minor became the first Brave since John Rocker in 2001 to challenge the team in an arbitration hearing. He won and was awarded $5.6 million. On March 3, 2015, Minor again began feeling tightness in his shoulder and was diagnosed with rotator cuff inflammation. As a result, he was placed on the disabled list on March 10. Minor attempted to start throwing programs in mid-March and early April, but felt discomfort both times. He was moved to the 60-day disabled list on May 2. Eleven days later, Minor underwent surgery for a torn labrum, and missed the rest of the season.
Minor became a free agent on December 2, 2015, when the Braves opted not to tender him a contract.
Minor signed a two-year contract with the Kansas City Royals worth $7.25 million with a club option worth $10 million for the 2018 season on February 19, 2016.
Minor began the 2016 season on the 60-day disabled list, still recovering from his previous shoulder surgery, and did not throw a pitch in the Majors for the entire year.
Minor leads with a four-seam fastball at 89–93 mph, a pitch he throws more than half the time. In relatively equal amounts he throws a slider (83–86), a circle change (82–84), and a knuckle curve (77–80). Left-handed hitters rarely see the changeup, especially with 2 strikes. The curve has a whiff rate of 39% over Minor's career. He has produced one of the league's lowest ground ball/fly ball ratios since he debuted.