Boston Red Sox Minor leagues Cuevas was signed by the
Boston Red Sox as an international free agent on June 2, 2009. He spent two-and-a-half seasons with the
DSL Red Sox spanning 2009–2011, before joining the
GCL rookie team late in 2011. He then was promoted to the
Low–A Lowell Spinners in 2012, and played with them for the entire season. Cuevas emerged during his first season with Lowell, as he posted eight wins and two losses in 77⅓
innings of work,
striking out 72 batters while
walking only 15 for a solid 4.80
SO/BB ratio. He also set a Spinners record for the lowest
earned run average (ERA) in a regular season, as his 1.40 ERA bested the 1.50 mark set by
Kyle Weiland in 2008. In addition, Cuevas ranked in the top of several statistical categories in the
New York–Penn League, tying for first in wins, for second both in ERA and innings, and for third in strikeouts. Cuevas was promoted to the
High–A Salem Red Sox in 2013 but struggled for most of the year. He posted an 8–9 record and a 5.05 ERA in 26 starts, while striking out 109 and walking 40 in 135⅓ innings. Overall, he was 8–5 with a 3.40 ERA in 19 starts for Portland, including 91 strikeouts and 41 walks in 95⅓ innings. He then joined the
Triple-A Pawtucket Red Sox in early August,
Major leagues Cuevas started the 2016 season with Triple-A Pawtucket. His contract was purchased by the Red Sox and he was added to their
40-man roster on April 20. Cuevas made his MLB debut the next day, pitching innings of relief against the
Tampa Bay Rays, giving up three hits and two runs while taking the loss. He made two other appearances with Boston during the season, both in June. Overall, with the
2016 Red Sox, Cuevas made three appearances with a 3.60 ERA in five innings pitched, striking out three and walking six. He also pitched 25 games (18 starts) with Pawtucket, compiling a 6–8 record with 4.19 ERA in 131 innings pitched. He became a
free agent after the season on November 7.
Detroit Tigers On November 19, 2016, Cuevas signed a minor league contract with the
Detroit Tigers. The Tigers purchased Cuevas's contract on April 13, 2017, adding him to their active roster. He made his only appearance with the Tigers the next day, pitching of an inning against the
Cleveland Indians, allowing four runs on three hits. He declined an outright assignment to Triple-A on June 1, 2017, and elected free agency.
Miami Marlins On June 7, 2017, Cuevas signed a minor league deal with the
Miami Marlins. He played for the Triple-A
New Orleans Baby Cakes, appearing in 15 games (11 starts) with a 2–7 record and 5.43 ERA. Cuevas elected free agency following the season on November 6.
Boston Red Sox (second stint) On January 5, 2018, Cuevas signed a minor league contract with the Red Sox. He was assigned to Triple-A and made 15 appearances (all starts) with Pawtucket through July 1, compiling a 5–5 record with 3.65 ERA in innings pitched. The Red Sox selected his contract on July 2, adding him to their major league roster. he was optioned back to Triple-A the following day. Boston activated Cuevas on August 11, as their 26th player for a
doubleheader against the
Baltimore Orioles; he pitched in relief in the second game, and was returned to Triple-A the following day. Cuevas was called up to Boston on September 1, when
rosters expanded. Cuevas finished the season with an 0–2 record and 7.41 ERA with Boston, while making nine appearances (one start) and striking out 20 in 17 innings. Cuevas was not included on Boston's postseason roster.
KT Wiz On November 20, 2018, Cuevas was granted his unconditional release so he could sign with the
KT Wiz of Korea's
KBO League. Cuevas produced a 13–10 record with a 3.62 ERA and 135 strikeouts over 184 innings in 2019. He re-signed with KT for the 2020 season for $900,000 (and $100,000 in incentives). On December 13, 2020, Cuevas re-signed with the Wiz for the 2021 season on a one-year $750K contract with a mutual option for the 2022 season. In November 2021, Cuevas was designated as the starting pitcher for the first game of the Korean Series in franchise history. He earned the win, pitching innings and giving up one run on seven hits, with eight strikeouts. He was named MVP of the game. On December 30, 2021, Cuevas re-signed with the Wiz for the 2022 season on a one-year contract worth up to $1.1 million. He suffered an elbow injury following his second start of the season, and was later released by the club on May 18, 2022.
Diablos Rojos del México On July 14, 2022, Cuevas signed with the
Diablos Rojos del México of the
Mexican League. He made one appearance for the Diablos in 2022, surrendering 4 earned runs in innings of work.
Los Angeles Dodgers On April 3, 2023, Cuevas signed a minor league contract with the
Los Angeles Dodgers organization. Cuevas made 11 appearances (nine starts) for the Triple–A
Oklahoma City Dodgers, registering a 6.14 ERA with 43 strikeouts in 44 innings of work.
KT Wiz (second stint) On June 9, 2023, Cuevas was signed by the
KT Wiz of the
KBO League. Cuevas was voted Player of the Month for August 2023, going 5–0 in five starts that month, giving up only two earned runs in 36 innings, for an ERA of 0.50. For the month of August 2023, he led all qualified KBO starters in wins and ERA, and was ranked second in strikeouts. In 18 starts for the team in 2023, he registered a 12–0 record and 2.60 ERA with 100 strikeouts across innings of work. On December 7, 2023, Cuevas re-signed with the Wiz on a one-year, $1.5 million contract. He made 31 starts for the team in 2024, compiling a 7–12 record and 4.10 ERA with 154 strikeouts across innings pitched. On November 29, 2024, Cuevas re–signed with the Wiz on a $1.5 million contract. He made 18 starts for the team in 2025, posting a 3–10 record and 5.40 ERA with 60 strikeouts across innings pitched. Cuevas was released by the Wiz following the signing of
Patrick Murphy on July 12, 2025.
CTBC Brothers On July 26, 2025, Cuevas signed with the
CTBC Brothers of the
Chinese Professional Baseball League; he and
Jesús Vargas were signed in an attempt to mitigate multiple injuries to other foreign players. In three starts for CTBC, he struggled to an 0–1 record and 15.26 ERA with seven strikeouts across innings pitched. Cuevas was released by the Brothers on August 31. ==International career==