Left-handed pitcher Henry Owens attended Edison High School in Huntington Beach, California. He earned the Cal-Hi Sports Mr. Baseball State Player of the Year Award in 2011, after posting a 12–1 record with a 1.15 earned run average (ERA) and three saves, while striking out 140 batters in 91⅔ innings of work. He also was named a 2011 Louisville Slugger High School All-American and earned 2011 Sunset League Pitcher of the Year honors for the two consecutive years.

The Boston Red Sox selected Owens in the first round (36th overall) of the 2011 MLB Draft. He signed with them on August 15, 2011 for a $1,500,000 signing bonus. Owens made his professional debut in 2012 with the Greenville Drive of the Class A South Atlantic League (SAL), sporting a 12–5 record and a 4.87 ERA in 23 games, including 130 strikeouts and 47 walks in 101⅔ innings. In addition, he finished fourth in the SAL for the most strikeouts, while tying for fourth in wins.

Entering 2013, Owens was ranked as the eighth best prospect in the Red Sox organization and ranked 77th in the MLB.com Top 100 Prospects list. He was promoted to the Salem Red Sox of the Class A-Advanced Carolina League on April 2. On July 17, he struck out a season-high tying 10 in six innings, while combining with two other pitchers to deliver the first no-hitter in franchise history in a 6–0 win over the Frederick Keys. Overall, he posted an 8–5 record and a 2.92 ERA over 20 starts for Salem, including a hitless streak of 19⅓ innings, gaining a promotion to the Portland Sea Dogs of the Class AA Eastern League on August 1.

With the Sea Dogs, Owens went 3–1 with a 1.78 ERA, striking out 46 in just 30⅓ innings over six starts. He was a Carolina League Mid-Season All-Star, as well as SoxProspects.com's 2013 Pitcher of the Year. Additionally, he made the Baseball America Minor League All-Star Team. Owens started the 2014 season by pitching a rain-shortened six-inning no-hitter for the Sea Dogs. Making 20 starts in Portland, Owens compiled a record of 14–4 with three complete games in 121 innings, with 126 strikeouts and an earned run average of 2.60. He was promoted to the Pawtucket Red Sox of the Class AAA International League on August 2, 2014. He was a mid-season and postseason All-Star with Portland as well as Eastern League Pitcher of the Year. Owens was invited to Spring Training with the Red Sox in 2015 but began the season with Pawtucket. He was 3–8 with a 3.16 ERA in 21 AAA starts.

The Red Sox promoted Owens to the major leagues on August 4. He made his debut by striking out the first batter he faced in the game, Jacoby Ellsbury and went on to allow three runs in five innings to take the loss against the New York Yankees. He collected his first MLB win in his second start on August 9 against the Detroit Tigers at Comerica Park, allowing one run through five innings. He made 11 starts in 2015, with a 4–4 record and 4.57 ERA. He spent most of the 2016 season with Pawtucket, where he was 10–7 with a 3.92 ERA in 24 starts but he also made five starts for the Red Sox in the majors, and was 0–2 with a 6.95 ERA in those appearances.