Jack Allan Westrup was the second of the three sons of George Westrup, insurance clerk, of Dulwich, and his wife, Harriet Sophia née Allan. He was educated at Dulwich College, London 1917-22, and at Balliol College, Oxford. He first read classics in which he gained first class honours in moderations (1924) and second class honours in literae humaniores (1926). He gained his B.Mus. degree in 1926, and a Master of Arts in 1929. He took an active part in music in the university as a keyboard and brass player. With an Italian expatriate Arundel del Re, he co-founded the Oxford University Opera Club while still an undergraduate, and was later its conductor. The club had a policy of producing works in English and used its funds to hire professional singers and conductors. In 1925, with William Henry Harris, he staged the first complete performance in modern times of Claudio Monteverdi's opera L'Orfeo (but only over certain obstacles presented by Sir Hugh Allen), and in 1927 he produced the first British performance of Monteverdi's L'incoronazione di Poppea. (In February 1975 he was present at a new production of Orfeo to celebrate the 50th anniversary of the University Opera Club.

He was a music critic for The Daily Telegraph 1934-39, and from also editor of the Monthly Musical Record 1933-45.

He gave classes at the Royal Academy of Music in London 1938-40. He was lecturer in music at King's College, Newcastle upon Tyne 1941-44, the Peyton and Barber Professor of Music at the University of Birmingham 1944-47, and Wadham College, Oxford 1947-71, where he held the Heather professorship. His students there included Joseph Horowitz, Alan Blyth and many others.

In 1947 he was named chairman of the editorial board of The New Oxford History of Music. In 1950 Oxford University allowed music to become an honours course for the first time, and Westrup was mainly instrumental in designing a new syllabus which demanded a wider knowledge of musical scholarship than the old B.Mus. That same year he conducted an edited version of Hector Berlioz's The Trojans with the Oxford University Opera Club, some passages from which were (non-commercially) recorded. In 1951 he was a co-founder and trustee of Musica Britannica, an authoritative national collection of British music. In 1952 he revised Ernest Walker's History of Music in England. In 1959 he succeeded Eric Blom as editor of Music & Letters. He was president of the Royal Musical Association 1958-63, the Incorporated Society of Musicians 1963, and the Royal College of Organists 1964-66.He conducted the Oxford Opera Club 1947-62, the Oxford University Orchestra 1954-63, and the Oxford Bach Choir and Oxford Orchestra Society 1970-71.

In 1966 he was one of the first advisers to Répertoire International de Littérature Musicale (RILM).

As late as 1968, he rejected an application by a student to undertake doctoral research on recordings of music, considering it not an appropriate topic for research.

Sir Jack Westrup died in 1975. His wife Solweig Maria née Rösell died in 1984.

He wrote a Divertimento in three short movements, for tenor saxophone, cello and piano. He also arranged a number of chorales by Johann Sebastian Bach for two pianos.

In 1946 Jack Westrup received an honorary degree of D.Mus. from Oxford University. He was knighted in 1961. His collection of 4,500 books on music history and musicology became the basis of the Westrup Library at the Guildhall School of Music and Drama.Oxford University Press now awards a Jack Westrup Prize in Musicology.