[2] During his senior season at Twin Valley, Alexy went 9 innings in which he threw 164 pitches, leading his team to a 2–1 win.

After signing, Alexy was assigned to the Arizona League Dodgers to make his professional debut; in 13+2⁄3 innings pitched for them, he posted a 1–0 record and a 4.61 ERA.

[5] On July 31, 2017, the Dodgers traded Alexy, along with Willie Calhoun and Brendon Davis, to the Texas Rangers in exchange for Yu Darvish.

He suffered a strained Latissimus dorsi muscle in his back during a game on May 1 and received a platelet rich plasma injection as part of his rehab process.

[14] Alexy did not play in a game in 2020 due to the cancellation of the Minor League Baseball season because of the COVID-19 pandemic.

[16] After going 3–1 with a 1.61 ERA and 57 strikeouts over 50+1⁄3 innings for Frisco, he was promoted to the Round Rock Express of the Triple-A West on August 3.

[17][18] On August 30, 2021, Alexy was promoted to the active roster to make his MLB debut that night versus the Colorado Rockies.

[25] On January 10, 2023, Alexy was traded to the Minnesota Twins in exchange for minor league right-hander Cristian Jimenez.

[31] On August 8, 2023, Alexy signed with the Southern Maryland Blue Crabs of the Atlantic League of Professional Baseball.