Allen Sapp was born in Philadelphia on December 10, 1922. He began piano lessons at an early age, developed an interest in music composition, and began performing his own work at the age of ten. After graduating from the Haverford School in 1939, he entered Harvard University, where he studied with Walter Piston, Archibald T. Davison, and Irving Fine. Many of Sapp's works were performed while he was an undergraduate, including the Andante for Orchestra, which won second prize in the New York Philharmonic Young Composer's Contest, and was performed by that group in April 1942.

Also in 1942, Sapp graduated from Harvard and was awarded the John Knowles Paine Fellowship, which allowed him to attend Tanglewood and to begin private composition lessons with Aaron Copland and Nadia Boulanger. After serving as a cryptanalyst in the U.S. Army during W.W.II, Sapp returned to Harvard and earned a Master's degree in 1949. He then joined the faculty at Harvard, teaching there until 1958, and at Wellesley College from 1958-61.

From 1961 to 1975 Sapp was chair of the Music Department of the University at Buffalo, The State University of New York. During this time, he helped establish the university as a leading center for experimental and new music, co-founding the Center of the Creative and Performing Arts with Lukas Foss in 1964. Sapp also assumed leadership positions in several national arts organizations, including Arts/Worth and the American Council for the Arts in Education, of which he was elected executive director in 1972.

Sapp later served as Provost for Communications and the Arts at Florida State University from 1976-78; and as Dean, then Professor of Composition, at the University of Cincinnati College-Conservatory of Music from 1978-93. In 1986 he was awarded the George Rieveschl Award for Distinguished Creative and Scholarly Works at the University of Cincinnati, and in 1993 was commissioned by the Cincinnati Chamber Orchestra to write a commemorative work for the group's twentieth anniversary. His works have also been performed by the Philharmonia Orchestra of the University of Cincinnati College-Conservatory of Music, the Cincinnati Wind Ensemble, the Buffalo Philharmonic, the Ohio University Chamber Orchestra, l'Orchestre de la Radiodiffusion Française, and the Boston Fine Arts Chamber orchestra.