Absil was born in Bonsecours, Hainaut, Belgium. His teacher there was Alphonse Oeyen, organist at the basilica of Bonsecours. From 1913 he studied organ and harmony at the Brussels Conservatoire, but upon graduating, decided to concentrate on composition instead.
In 1922 Absil won the Belgian Prix de Rome and in 1934 the Prix Rubens, which allowed him to travel to Paris. Here, he met fellow contemporary composers Ibert, Milhaud, and Honegger. Absil gained international prominence with the premiere of his first piano concerto (op. 30), composed for the 1938 Queen Elizabeth Competition for Piano (Ysaye), for which it was the compulsory piece for all finalists. Only one of those, Moura Lympany, who won second prize (after Emil Gilels), performed the piece entirely from memory mistake free.
From 1930 onwards, Absil taught harmony at the Brussels Conservatoire, becoming a professor of counterpoint there six years later. Amongst his Conservatoire pupils was Paul Danblon. He also taught at Chapelle Musicale Reine Elisabeth and the Etterbeek Music School. During this period also, he was, with Charles Leirens, the first editor of the Revue Internationale de Musique (1936–1952). From 1955 he served as a member of Belgium's Royal Academy. He died in Brussels at the age of 80.