On leaving school he worked part-time as a reporter for the Irish Times, part-time as an actor at the Abbey Theatre, Dublin, before becoming a professional actor for the Gate Theatre. In World War II he joined the Forces entertainment organisation ENSA.

His television career began during the 1950s, when he was cast in small roles in three Nigel Kneale / Rudolph Cartier productions for BBC Television: as a drunk in The Quatermass Experiment (1953), as both an old man in a pub and later a prisoner in Nineteen Eighty-Four (1954) and as a tramp in Quatermass II (1955). All of these roles earned him a reputation for playing old men, though he was only at the time in his forties.

It was this ability to play old men that led to his casting in his most famous role, as Albert Steptoe, the irascible father in Steptoe and Son. Initially this was a pilot on the BBC's Comedy Playhouse anthology strand: but its success led to a full series being commissioned, which lasted throughout the 1960s and into the 1970s. There were also two feature film spin-offs, a stage show and an American re-make entitled Sanford and Son, based on the original British scripts. In the latter, Brambell's part was taken by Red Foxx.

The success of Steptoe and Son made Brambell a high profile figure on British television, and earned him the major role of Paul McCartney's grandfather in The Beatles' first film, A Hard Day's Night. A running joke is made throughout the film of his character being "a very clean old man." This is in reference to his on-screen son, Harold, in Steptoe and Son constantly referring to his father as "you dirty old man!" (In real life too, he was nothing like his Steptoe persona, being dapper and well-spoken).

In 2002, Channel 4 broadcast a documentary film on the off-screen life of Brambell and his relationship with Harry H. Corbett, who played Harold Steptoe in Steptoe and Son. The film revealed that the two men detested each other and were barely on speaking terms outside of takes by the end of the programme's run. In a series almost entirely based around the pair of them with no other regular characters, this made production of the series difficult and stressful. This tension partly related to Brambell's difficult private life. He was alcoholic, which made him forget his lines and caused other problems both on and off the set. He was also a homosexual at a time when it was almost impossible for public figures to be openly gay. Brambell was arrested and charged for 'cottaging' in the early 1960s and subsequently holidayed annually in Asia. Earlier in his life he had been married, from 1948 to 1955, to Molly Josephine but the relationship ended after she gave birth to the child of their lodger, Roderick Fisher in 1953.

After the final series of Steptoe and Son was made in 1974, Brambell had some guest roles in films and on television, but both he and Corbett found themselves heavily type cast as their famous characters. In an attempt to take advantage of this situation, they undertook a tour of Australia in the late 1970s with a Steptoe and Son stage show: however, with the pair openly despising each other, the tour was a disaster and a working relationship proved impossible. On one occasion, Brambell used bad language and was openly derogatory about the Australian people in an interview. Brambell did, however, appear on the BBC's television news to pay tribute to Corbett after the latter's death from a heart attack in 1982. The following year Brambell appeared in Terence Davies' film Death and Transfiguration, playing a dying elderly man who finally comes to terms with his homosexuality.

Brambell himself died in London less than three years later, of cancer. He was seventy-two. News of his death received far less attention than that of his co-star, and his funeral was sparsely attended. He left £170,000 to his partner.