On June 23, 2011, Brazobán signed with the Colorado Rockies as an international free agent.[1] He made his professional debut with the Dominican Summer League Rockies in 2012, making 18 appearances and registering a 3.71 ERA with 15 strikeouts and 2 saves in 17.0 innings pitched.[2] Brazobán made 21 appearances for the Low-A Tri-City Dust Devils in 2013, struggling to a 7.78 ERA with 24 strikeouts in 19+2⁄3 innings of work.[3] He split the 2014 season between Tri-City and the High-A Modesto Nuts, pitching in 23 games and logging a 3.38 ERA with 28 strikeouts in 34+2⁄3 innings pitched.[4]
Brazobán missed all of 2015 with an injury, and elected free agency following the season on November 6, 2015.[5] After being out of affiliated baseball in 2016, Brazobán re-signed with the Rockies organization on a minor league contract on November 18, 2016.[6] In 2017, he made 37 appearances split between the High-A Lancaster JetHawks and Double-A Hartford Yard Goats, posting a cumulative 5.77 ERA with 32 strikeouts and 5 saves in 39.0 innings of work. Brazobán elected free agency following the season on November 6, 2017.[7]
He did not play for the team in 2019, after being unable to report to the team due to visa issues. On February 5, 2020, Brazobán re-signed with the Barnstormers.[10] He did not play in a game in 2020 due to the cancellation of the Atlantic League season because of the COVID-19 pandemic.[11]
On January 4, 2022, Brazoban signed a contract with the Miami Marlins organization, where he was assigned to their Triple-A affiliate, Jacksonville Jumbo Shrimp.[14] In 27 games, he recorded a 3.18 ERA with 59 strikeouts in 45+1⁄3 innings pitched.[15] On July 24, Brazobán was selected to the 40-man roster and promoted to the major leagues for the first time. He made his MLB debut against the Pittsburgh Pirates that day at the age of 32.[16] He threw one scoreless inning, striking out his first batter faced, Oneil Cruz.[17] In his rookie campaign, Brazobán logged a 3.09 ERA with 40 strikeouts in 27 appearances out of the bullpen.[18]
In 2023, Brazobán made 50 appearances out of the bullpen for Miami, registering a 4.14 ERA with 65 strikeouts across 58+2⁄3 innings pitched.[19] Brazobán missed spring training in 2024 after a visa issue prevented him from entering the United States.[20] As a result of the issue persisting into the regular season, the Marlins placed him on the restricted list to begin the year.[21] Brazobán was activated on May 7, 2024, and was subsequently optioned to Triple–A Jacksonville.[22] In 20 appearances for Miami, he logged a 2.93 ERA with 34 strikeouts across 30+2⁄3 innings pitched.
New York Mets
On July 30, 2024, the Marlins traded Brazobán to the New York Mets in exchange for minor league infielder Wilfredo Lara.[23] Brazobán made 19 appearances for New York in 2024, posting a 5.14 ERA with 17 strikeouts across 21 innings pitched.[24]