Balazovic spent his first two professional seasons in 2016 and 2017 with the Gulf Coast Twins, posting a 1.97 ERA and a 4.91 ERA, respectively. He spent 2018 with the Cedar Rapids Kernels, going 7–3 with a 3.94 ERA in 12 games (11 starts),[3] and returned there to begin 2019 before being promoted to the Fort Myers Miracle.[4][5] In July, he represented the Twins in the 2019 All-Star Futures Game.[6] Over 19 games (18 starts) with Cedar Rapids and Fort Myers, he pitched to an 8–5 record and a 2.69 ERA, striking out 129 over 93+2⁄3 innings. In July and August he played for team Canada in the 2019 Pan American Games.[7]
Balazovic did not play in a game in 2020 due to the cancellation of the minor league season because of the COVID-19 pandemic.[8] The Twins added Balazovic to their 40-man roster after the 2020 season.[9]
Balazovic spent the 2021 season with the Double-A Wichita Wind Surge, recording a 5–4 record and 3.62 ERA with 102 strikeouts in 97.0 innings pitched across 20 starts. He pitched in 22 games (21 starts) for the Triple-A St. Paul Saints in 2022, but struggled to an 0–7 record and 7.39 ERA with 76 strikeouts in 70.2 innings pitched.
On February 11, 2023, Balazovic suffered a broken jaw after being struck twice by a stranger and had his jaw wired shut.[10] On March 3, Balazovic was optioned to Triple-A St. Paul to begin the season.[11] He made 14 appearances, mainly working out of the bullpen,[12] and logged a 4.79 ERA with 50 strikeouts in 35+2⁄3 innings of work. On June 18, Balazovic was promoted to the major leagues for the first time after Jorge Lopez was placed on the injured list.[13] In the major leagues, Balazovic recorded a 1–0 record and a 4.44 ERA with 17 strikeouts in 24.1 innings pitched over 18 games. He was designated for assignment on February 7, 2024.[14] He cleared waivers and was sent outright to Triple-A on February 13.
Doosan Bears
On July 4, 2024, Balazovic signed with the Doosan Bears of the KBO League.[15][16] He made 12 starts for the Bears, posting a 2–6 record with a 4.26 ERA and 69 strikeouts across 57 innings pitched. Balazovic became a free agent following the season.
Detroit Tigers
On December 22, 2024, Balazovic signed a minor league contract with the Detroit Tigers.[17]