Askenase was born in Lemberg. He studied with Theodor Pollak, a professor and director of the Ludwik Marek School of Music in Lemberg. Then with Emil von Sauer, a pupil of Liszt, at the Vienna Academy of Music. In 1919 he made his debut in Vienna, and subsequently toured throughout the world. He also taught at the Rotterdam Conservatory of Music from 1937 to 1940 and the Brussels Conservatory of Music from 1954 to 1961. In 1950 he became a naturalized Belgian citizen. He recorded extensively the works of Chopin for the Deutsche Grammophon label in the 1950s and 1960s.