Asencio began composing in the early 1920s, and his first major success was the ballet Fuego de Fiesta (1926).
His earlier music displays an evident influence of the composer Manuel de Falla.
In 1934 he helped formed the Grupo de los Jóvenes, a group of talented young musicians in Valencia whose members also included Vicente Garcés and Ricardo Olmos among others.
Asencio's best-known works for the guitar are Elegía a Manuel de Falla (1946), Sonatina (1949), Collectici Íntim (1965), and Dipso (1973).
He also wrote music for several successful ballets, most notably La casada infiel (1949) which is considered his best work in that genre.