After completing his education at the Secondary Grammar School Franz-Josef, he attended the Academy of Music in Budapest, where he studied with Leo Weiner.

During World War II and the Occupation of France, Kosma was placed under house arrest in the Alpes-Maritimes region and was banned from composition.

Under this arrangement, he wrote the "pantomime" of the music for Les Enfants du Paradis (1945), made under the occupation but released after the liberation.

Among his other credits are the scores to Voyage Surprise (1946) and Le Testament du docteur Cordelier (The Doctor's Horrible Experiment, 1959), the last of which was made for television.

He was also known for writing the standard classical-jazz piece "Les feuilles mortes" ("Autumn Leaves"), with French lyrics by Prévert and later English lyrics by Johnny Mercer, which was derived from music in Marcel Carné's film Les Portes de la Nuit (1946).