The Detroit Tigers selected Green in the 11th round of the 2013 MLB draft. He made his professional debut that year with the GCL Tigers, and after two games, was promoted to the Lakeland Flying Tigers where he finished the year with a 3-0 record and a 3.63 ERA in 17.1 innings pitched. He pitched for the West Michigan Whitecaps in 2014 where he was 6-4 with a 3.11 ERA in 23 starts. In 2015, he played for the Erie SeaWolves where he compiled a 5-14 record with a 5.93 ERA in 27 starts.
On December 9, 2015, the Tigers traded Green and Luis Cessa to the New York Yankees for Justin Wilson. He received a non-roster spring training invitation on February 5, 2016. He began the year with the Scranton/Wilkes-Barre RailRiders. The Yankees promoted Green to the major leagues on May 14, 2016. He made his major league debut on May 16. After being called up as a reliever, the Yankees shifted Green to the rotation. After four starts, the Yankees shut down Green for the season after he suffered a sprained elbow ligament. Green appeared in 12 games, with a 2–4 record, 4.73 ERA and 52 strikeouts in innings. At the end of spring training in 2017, the Yankees optioned Green to the RailRiders. On May 8, Green was called up to the Yankees. Green made his season debut with the Yankees on May 9 and was used primarily as a reliever for the season. On June 11, Green made his first start of season, a spot start, against the Baltimore Orioles. He allowed two runs in two innings, striking out three. Green ended the season with a 5–0 record, 1.83 ERA and 103 strikeouts in 69 innings. He and Dellin Betances became the sixth pair of teammates to strike out 100 batters as a reliever. In the first inning of the 2017 American League Wild Card Game, Green relieved Luis Severino with two runners on base after three runners had scored. Green ended the inning with consecutive strikeouts and later pitched a scoreless second inning before being taken out in the third inning. In two innings pitched, Green allowed one earned run and struck out four. He finished the 2017 season with a 5–0 record, a 1.83 ERA, and a 0.74 WHIP in 40 games. In 2018, in which he was 8–3, balls hit against him had the highest average exit velocity in the major leagues, at 91.0 miles per hour. Green struggled in April 2019, allowing 14 earned runs in innings pitched. The Yankees optioned Green to Scranton/Wilkes-Barre on April 24. He made his return on May 12, recording three strikeouts in the ninth inning to close out a 7–1 win against Tampa Bay Rays. For the 2019 season, Green was 4–4 with a 4.17 ERA.