Born in Moson in Hungary, Flesch studied at the conservatoires in Vienna and Paris. He settled in Berlin, and as well as being known for his solo performances in a very wide range of repertoire (from Baroque music to contemporary), gained fame as a chamber music performer and as a violin pedagogue. He published a number instructional books including Die Kunst des Violin-Spiels (1923). Among his pupils were Ida Haendel and Henryk Szeryng, and he was consulted by Louis Krasner over technical difficulties in the Violin Concerto by Alban Berg which Krasner was to premiere.
Flesch owned the Brancaccio Stradivarius, but had to sell it in 1928 after losing all his money on the New York Stock Exchange.
Flesch died in Lucerne.