Winn spent his freshman, sophomore, and junior years of high school at Silver Creek High School in Longmont, Colorado.[1] As a junior in 2017 he was 9–0 with a 0.73 ERA with 95 strikeouts in 47+2⁄3 innings, along with batting .388 with two home runs and 19 RBIs, earning himself the title of Colorado's Gatorade Baseball Player of the Year.[2] That summer, in July, he played in the Under Armour All-America Baseball Game at Wrigley Field.[3]
The Texas Rangers selected Winn in the first round, with the 15th overall selection, of the 2018 Major League Baseball draft and he signed with the Rangers for a $3.15 million signing bonus.[10][11] After signing, Winn did not appear in an official game with a Rangers' affiliate in the 2018 season. Instead, he took part in a new program put in place by Texas for their newly drafted high school pitchers. The "de-load" program, as the organization calls it, emphasizes building a foundation mentally and physically, while resting bodies from a strenuous senior season and pre-draft showcase circuit. The players are put through a strength program and classroom work until post-season fall instructional training starts.[12] Winn was ranked as the #89 overall prospect in baseball by MLB Pipeline in their preseason 2019 Top 100 list.[13] He was also ranked as the #72 overall prospect in baseball by ESPN's Keith Law in his preseason 2019 Top 100 list.[14] He was ranked as the #78 overall prospect in baseball by Fangraphs in their preseason 2019 Top 130 list.[15]
Winn made his professional debut in 2019 when was assigned to the Hickory Crawdads of the Single–ASouth Atlantic League with whom he started 18 games, compiling a 4–4 record with a 4.46 ERA over 68+2⁄3 innings.[16][17][18] Winn spent 2020 at the Rangers Alternate Training Site, due to the cancellation of the season because of the COVID-19 pandemic.[19] To begin the 2021 season, he was assigned to the Frisco RoughRiders of the Double-ACentral League.[20] In June, Winn was selected to play in the All-Star Futures Game at Coors Field.[21] In 19 games for Frisco in 2021, Winn posted a 3–3 record with a 2.31 ERA and 97 strikeouts over 78 innings.[22] After the end of Frisco's season in mid-September, he was promoted to the Round Rock Express of the Triple-A West where he posted a 3.38 ERA over two outings.[23][24] He was named the Rangers' 2021 Nolan Ryan Pitcher of the Year.[25][26] Winn spent the 2022 season back with Round Rock.[27] Over 28 starts, he went 9–8 with a 6.51 ERA, 123 strikeouts, and 87 walks (the most in the minor leagues) over 121+2⁄3 innings.[28][29]
On November 15, 2022, the Rangers selected Winn to their 40-man roster to protect him from the Rule 5 draft.[30] Winn returned once again to Round Rock for the 2023 season, struggling to a 9–8 record with a 7.22 ERA and 97 strikeouts over 101 innings.[31][32] Following that season, he played for the Criollos de Caguas of the Puerto Rican Winter League, going 2–0 with a 2.63 ERA and 24 strikeouts over 24 innings.[33]
Winn was optioned to Triple–A Round Rock to begin the 2024 season.[34] On April 14, 2024, Winn was promoted to the major leagues for the first time.[35] After struggling to a 7.79 ERA in 13 games, he was placed on the injured list with right shoulder discomfort on June 24.[36] Winn was transferred to the 60–day injured list two days later.[37]