Rosset turned professional in 1988 and won his first tour singles title in 1989 in Geneva as a wildcard, defeating Guillermo Pérez Roldán.

Representing Switzerland at the Olympic Games in Barcelona, he defeated several top players en route to qualifying for the men's singles final, including Jim Courier, Goran Ivanišević, Wayne Ferreira, and Emilio Sánchez.

Switzerland lost in the final to the United States despite Rosset's winning a five-set singles rubber against Jim Courier (who was ranked world No.

Rosset's most memorable Davis Cup match came in defeat in a singles rubber against Arnaud Clément of France in 2001, which he lost 15–13 in the fifth set after 5 hours and 46 minutes.

That year he also achieved his best performance at a Grand Slam, the 1996 French Open when he defeated Carl-Uwe Steeb, Jiří Novák, Jakob Hlasek, Stefan Edberg and Bernd Karbacher before losing to Michael Stich in the semifinals.