Gasquet was the third of six children of Raymond Gasquet, a physician whose French naval officer father had emigrated to England during the British evacuation of Toulon in 1793. His mother was a Yorkshirewoman. He was born at 26 Euston Place, Somers Town, London.
Educated at Downside School, he entered the Benedictines in 1865 at Belmont Priory. He moved to Downside Abbey where he was professed and, on 19 December 1871, ordained a priest. From 1878 to 1885 he was prior of Downside Abbey, resigning because of ill health.
Upon his recovery he became a member of the Pontifical Commission to study the validity of the Anglican ordinations(1896) leading to Apostolicae curae, to which his historical contribution was major. In 1900, he became abbot president of the English Benedictines. He was President of the Pontifical Commission for Revision of the Vulgate, 1907. He also authored the major history of the Venerable English College at Rome.
He was created Cardinal-deacon in 1914 with the titular church of San Giorgio in Velabro. He was conferred the titular church of Santa Maria in Portico in 1915.
In 1917, he was appointed Archivist of the Vatican Secret Archives. In 1924, he was appointed Librarian of the Vatican Library.