After a teenage career in motocross, Barbazza began circuit racing in Formula Monza in 1982 before moving into the Italian Formula Three Championship the following year. He finished sixth in the series in 1984, then improved significantly in 1985, winning four races and finishing third in the championship. That success prompted a move to the United States, where he entered the American Racing Series. He won five races and took the title at his first attempt.
Building on his American Racing Series title, Barbazza entered CART in 1987. He finished third at the 1987 Indianapolis 500, earning recognition as CART's Rookie of the Year. Despite this performance, he was unable to secure a full-time CART seat for 1988 and made only two starts in Formula 3000 that year, failing to qualify on three occasions. He returned to CART in 1989 for eight races, posting a best finish of eighth at Toronto, and also contested eight Japanese Formula 3000 races during the period.
In 1990, Barbazza joined the Crypton team for a full season in International F3000. He finished 16th in the points standings, with a fourth place at his home circuit in Monza his only points-scoring result. For 1991, Barbazza signed with the struggling AGS Formula One team, joining from the third round of the championship as a replacement for Stefan Johansson. He failed to qualify for all twelve races he attempted with the team. AGS shut down after both Barbazza and his teammate Olivier Grouillard failed to pre-qualify for the 1991 Spanish Grand Prix.
He returned to CART for the 1992 season with the Arciero team, performing respectably in outdated equipment through the opening three races, but crashed his 1990-vintage Lola in practice for the 1992 Indianapolis 500 and was unable to qualify. That marked his final CART appearance.
In 1993, Barbazza rejoined Formula One with Minardi and scored championship points twice during his first four races with the team. However, he was replaced by Pierluigi Martini after eight races, ending his second and final Formula One stint.
In 1995, while racing a Ferrari 333SP sports prototype at the Road Atlanta circuit, Barbazza was involved in a major accident with Jeremy Dale. The crash resulted in severe head and chest injuries that left him in critical condition, in a coma, and on artificial respiration. He fully recovered but did not return to professional racing.
Following his recovery, Barbazza established a go-kart circuit in Monza and began designing crash barriers. He later relocated to Cuba, where he set up a fishing resort called La Villa Clara in the north of the country. He has raced at a local karting track in Cuba, rediscovering his enjoyment of motorsport in an informal setting.