The Boston Red Sox selected Kopech in the first round, with the 33rd overall selection, of the 2014 MLB Draft. He received a $1.6 million signing bonus from Boston. Kopech started his professional career with the rookie-level Gulf Coast League Red Sox in their 2014 season. He posted a 0–1 record and a 4.61 ERA in eight starts, which included 16 strikeouts and nine walks in 13⅓ innings pitched. In 2015, Kopech pitched for the Greenville Drive of the Class ASouth Atlantic League, where he was 4–5 with a 2.63 ERA in 15 games. He struck out 70 batters and walked 27, in 65 innings, while earning a selection to the SAL All-Star Game. On July 16, he was suspended without pay for 50 games after testing positive for Oxilofrine, which is a banned substance under the Minor League Drug Prevention and Treatment Program. In the 2016 spring training, Kopech broke his right hand during an altercation involving a roommate. He returned to action in the season-opener for the Lowell Spinners of the Class A-Short SeasonNew York-Penn League, and was promoted to the Salem Red Sox of the Class A-AdvancedCarolina League. In his first five starts at Salem, Kopech yielded only three earned runs in 29 innings with at least seven strikeouts in every game. In total, Kopech made 11 starts, going 4–1 with a 2.25 ERA across 52 innings. During that stretch, he struck out 82 hitters, the most for a pitcher in MiLB for the month of August. He posted a 2.08 ERA and gave only 29 walks, striking out double-digit batters in four of his last six starts, including a career-high 11 twice. After his solid pitching effort, Kopech was named the Carolina League Player of the Month for August. The award came along with his selection as the league's Pitcher of the Week for August 22–28. He carried a 0.93 ERA into his final start of the season before a tough outing against the Winston-Salem Dash on August 31. Overall, he collected 86 strikeouts in 56⅓ innings during the two stints, along with a 4–1 record and a 2.08 ERA. According to Baseball America, Kopech threw one of the fastest pitches in professional baseball history against the Wilmington Blue Rocks on July 13, 2016. Kopech sat at 98 mph in the game, and touched 100 mph and beyond on a regular basis, until a 105 mph pitch was double checked by different radar guns in the Salem ballpark. In an offseason workout on January 17, 2017, throwing from flat ground with run-up, Kopech threw a pitch that was unofficially clocked at 110 mph with a 3 oz ball.
On December 6, 2016, the Red Sox traded Kopech, Yoan Moncada, Luis Alexander Basabe, and Victor Diaz to the Chicago White Sox for Chris Sale. In 2017, Kopech was named the starter for the North Division in the Southern League All-Star Game after he went 4–3 with a 2.93 ERA over 58 1/3 innings and a league best 80 strikeouts in his first 11 starts for the Double-A Birmingham Barons. He was later named to the U.S. team of the 2017 All-Star Futures Game, striking out White Sox teammate Yoan Moncada of the World team. On August 18, 2017, the White Sox promoted Kopech to the Triple-A Charlotte Knights after he posted a 2.87 ERA over innings in 22 starts with 155 strikeouts. Over his final five starts in Birmingham, Kopech allowed two earned runs and accumulated 49 strikeouts. The White Sox promoted Kopech to the major leagues on August 21, and he made his debut that day. His outing lasted only two innings because of rain. He struck out four batters and did not allow any runs. On his second pitch of the game he gave up his first hit, a single, to Joe Mauer. At the time of his debut, he was considered the 13th best prospect in all of MLB. On September 7, 2018, it was announced that Kopech would undergo Tommy John surgery, putting him out for the rest of the 2018 season plus all of 2019 as well. He underwent successful Tommy John surgery on September 18. On July 10, 2020, Kopech announced he would be sitting out the 2020 season due to the COVID-19 pandemic.
Personal life
In 2019, Kopech announced his engagement to Canadian actress Vanessa Morgan. Kopech and Morgan were married on January 4, 2020. On July 24, 2020 Morgan, announced via Instagram that she was pregnant, with a boy due in January 2021. Days later, it was reported that Kopech had filed for divorce on July 26, 2020, and a spokeswoman for Morgan announced that he was the father of her baby.