Born in Colombes, not far from Paris, Cudini first fought his way through the Formula Renault junior series, winning the European Cup there in 1972. After a few more years in this formula, he competed with considerable success in the French Touring Car Championship from 1977 onwards, where, after occasional excursions into the World Sports Car Championship, he celebrated his greatest triumph in 1983, winning the title in an Alfa Romeo GTV6.
A first career highlight with Mercedes-Benz came in 1989 with 6th place in the 24 Hours of Le Mans with his French team-mates Jean-Louis Schlesser and Jean-Pierre Jabouille in the Sauber-Mercedes C 9. Despite his mature age, he also made a name for himself in the highly competitive German Touring Car Championship, where he was successful from 1988 to 1991 in a 190 E 2.3-16 DTM and its evolution models. He finished his debut season in fifth place in the overall standings, and in his final year in the DTM he took sixth place.
After that, the Frenchman turned to other touring car series, ended his career in 2000, but started again in the French GT Championship in 2005.