Quirk was born at Lambfell, in Michael on the Isle of Man, the son of Thomas and Amy Randolph Quirk. He attended King William's College on the Isle of Man and then went to University College London to read English under Albert Hugh Smith. His studies began in 1939 but were interrupted by the war in 1940, to be completed from 1945 to 1947. He was awarded MA, PhD, and D.Lit and became a lecturer in English at UCL from 1947 to 1954. He was also a Commonwealth Fund Harkness Fellow, at Yale and Michigan from 1951 to 1952. In 1954 he became Reader at the University of Durham, becoming Professor there in 1958. He returned to UCL as Professor in 1960 and in 1968 succeeded Smith as Quain Professor, a post he held until 1981.
Quirk lectured and taught seminars at University College, London, in Old English (Anglo-Saxon) and in History of the English Language. These two disciplines were part of a ten-discipline set of Final examinations in the undergraduate syllabus. At that time, Old and Middle English, along with History of the English Language, were all compulsory subjects. He also worked closely with A.C. Gimson and J.D. O'Connor of the Phonetics Department, sitting in as an examiner for Phonetics oral examinations on occasion.
During the early 1960s, Randolph Quirk and colleagues, among them Valerie Adams, Derek Davy and David Crystal, conducted an ambitious project known as the Survey of English Usage. This compilation of a large body of English language data (a corpus) comprised one million words as recorded in actual use in everyday life. Previous grammars had tended to overuse the canon of English Literature. The project was to be the foundation of A Comprehensive Grammar of the English Language, a reference grammar widely used around the world. This was the first grammar of English in real use rather than one based on rules handed down by teachers and scholars from Latin and Greek models. These had been considered to be "correct" models for living English. Instead of declaring what was correct grammatical usage, Quirk and his collaborators proposed a descriptive rather than prescriptive grammar, showing readers that different groups of English speakers chose different usages, arguing that what is correct is what communicates effectively. Basil Bernstein, sociolinguist, made his name from showing similar choices of variants of English usage.
One of Quirk's favourite enterprises was the "London University Summer School of English" where the above-mentioned colleagues and other budding leading scholars and friends of his came to teach for a month. It was considered the most eminent body of English teachers anywhere in the world. The resident students were foreign academics, teachers, and students. He threw himself into the social life with gusto and enjoyed singing Victorian ballads over a "couple of pints" with a Cockney accent. When the School moved away from Queen Elizabeth College to New Cross, numbers fell rapidly. The next and last successful Director was the phonetician J D O'Connor.
Quirk has been an open Labour supporter all his life. He was president of the British Academy from 1985 to 1989 and became a life peer as Baron Quirk, of Bloomsbury in the London Borough of Camden in 1994.
He currently resides in Germany and England, with his wife, German linguist Gabriele Stein.