Webb was born at Birkenhead, the son of John Racker Webb and his wife Elizabeth Hodgson Fairbank. He was educated at Birkenhead School and in 1917 volunteered to join the Royal Navy. He went to Magdalene College, Cambridge in 1919 where he read English, and graduated in 1921. When he moved to London, he met members of the Bloomsbury group, including Roger Fry.
Webb started to write articles on art for the Burlington Magazine. In 1929 he started lecturing at Cambridge and also became a lecturer at the Courtauld Institute of Art in 1934. In 1938 he became Slade Professor of Fine Art at Cambridge.
When World War II broke out, he joined the Navy again, working for Naval Intelligence at the Admiralty. He was then in the historical section of the War Cabinet Office and joined the Monuments, Fine Arts, and Archives program.
He resumed the Slade professorship in 1948, and then became secretary to the Royal Commission on Historical Monuments. He was also a member of the Royal Fine Arts Commission, and was elected a fellow of the British Academy in 1957. In 1956 he wrote a volume on medieval British architecture for the Pelican History of Art.
- The Complete Works of Sir John Vanbrugh. 4 vols. with Bonamy Dobrée, The Nonesuch Press, 1927–1928
- "Architecture and sculpture" in Roger Fry Georgian Art (1760–1820), B. T. Batsford, 1929
- The Letters and Drawings of Nicholas Hawksmoor Relating to the Building of the Mausoleum at Castle Howard, 1726–1742, Walpole Society 17 (1929)
- Gothic Architecture in England, British Council/Longmans, Green, 1951
- Baroque Art: Annual Lecture on Aspects of Art, Henriette Hertz Trust, 1947. British Academy, 1951
- Architecture in Britain: the Middle Ages, (Pelican History of Art 12), Penguin Books, 1956
- Dictionary of Art Historians
- Oxford Dictionary of National Biography, John Crittall, Webb, Geoffrey Fairbank (1898–1970) 2004