Jarnach was the son of a Spanish sculptor and a Flemish mother. Until 1914 he lived in Paris, where he studied piano under Édouard Risler and harmony under Albert Lavignac at the Conservatoire de Paris. During the First World War he was a student of Ferruccio Busoni in Zürich. He later completed the opera Doktor Faust which Busoni had left unfinished on his death in 1924.
In the 1920s Jarnach worked in Berlin as a pianist, conductor and composer. In 1927 he became a teacher in composition at the Hochschule für Musik Köln. In 1949 he founded the Hamburger Musikhochschule (Hamburg Music Academy) which he directed until 1959 and at which he taught until 1970. Some of his students include Kurt Weill, Otto Luening, Wilhelm Maler, Bernd Alois Zimmermann, Jürg Baur, Eberhard Werdin and Nikos Skalkottas.
Jarnach composed a Sinfonia brevis, a prelude for large orchestra, a quartet and a quintet for strings, further chamber music, especially for violin and piano, and vocal works.