Corbett, the youngest of seven children, was born in Rangoon, Burma, where his father, George Corbett (1885/86-1943), was serving as a company quartermaster sergeant in the South Staffordshire Regiment of the British Army, stationed at a cantonment as part of the Colonial defence forces. Corbett was sent to Britain after his mother, Caroline Emily, née Barnsley, (1884–1926) died of dysentery when he was eighteen months old. He was then brought up by his aunt, Annie Williams, in Earl Street, Ardwick, Manchester and later on a new council estate in Wythenshawe. He attended Ross Place and Benchill Primary Schools; although he passed the scholarship exam for entry to Chorlton Grammar School, he was not able to take up his place there and instead attended Sharston Secondary School.

Corbett married twice, first to the Actress Sheila Steafel, and then to Actress Maureen Blott (stage name Crombie) (1943–1999), with whom he had two children, Jonathan and Susannah. Susannah is an Actress and author, and she has written a biography of her father, Harry H. Corbett: The Front Legs of the Cow, which was published in March 2012.

Corbett enlisted in the Royal Marines during the Second World War, and served in the Home Fleet on the heavy cruiser HMS Devonshire. After VJ Day in 1945, he was posted to the Far East, where he was involved in quelling unrest in New Guinea and reportedly killed two Japanese Soldiers there whilst engaged in hand-to-hand fighting. He was then posted to Tonga, but deserted and remained in Australia before handing himself in to the Military Police. His military Service left him with a damaged bladder following an infection and a red mark on his eye caused by a thorn, which was not treated until late in his life.

Upon return to civilian life, Corbett trained as a radiographer before taking up acting as a career, initially in repertory. In the early 1950s, he added the initial "H" to avoid confusion with the television entertainer Harry Corbett, known for his act with the glove-puppet Sooty. He joked that "H" stood for "hennyfink", a Cockney pronunciation of "anything". In 1956, he appeared on stage in The Family Reunion at the Phoenix Theatre in London.

From 1958, he began to appear regularly in films, coming to public attention as a serious, intense performer, in contrast to his later reputation in sitcom. He appeared in television dramas such as The Adventures of Robin Hood (as four different characters in different episodes between 1957 and 1960) and Police Surgeon (1960). He also worked and studied Stanislavski's system at Joan Littlewood's Theatre Workshop at the Theatre Royal in Stratford, London.

Scriptwriters Galton and Simpson, who had been successful with Hancock's Half Hour, changed Corbett's life. In 1962, at their request, Corbett appeared in "The Offer", an episode of the BBC's anthology series of one-off comedy plays, Comedy Playhouse, written by Galton and Simpson. He played Harold Steptoe, a rag-and-bone man who lives with his irascible widower father, Albert (Wilfrid Brambell) in a dilapidated house attached to their junkyard and stable for their cart horse, Hercules. Corbett was, at the time, working at the Bristol Old Vic where he appeared as Macbeth.

Steptoe and Son led Corbett to comedy films: as James Ryder in Ladies Who Do (1963); with Ronnie Barker in The Bargee (1964), written by Galton and Simpson; Carry On Screaming! (1966); the "Lust" segment of The Magnificent Seven Deadly Sins (1971); and Terry Gilliam's Jabberwocky (1977). There were two Steptoe and Son films: Steptoe and Son (1972) and Steptoe and Son Ride Again (1973). He also had the leading role in two other television series, Mr. Aitch (written especially for him, 1967) and Grundy (1980).

Corbett was a Labour Party campaigner, appeared in a party political broadcast, and was a guest of Prime Minister Harold Wilson. The television character Harold Steptoe appears as the Labour Party secretary for Shepherd's Bush West in the sixth series episode, "Tea for Two". In 1969, Corbett appeared as Harold Steptoe in a Labour Party political broadcast, where Bob Mellish had to argue against Harold's accusation that all parties are the same. This was not in any way connected to Galton and Simpson who wrote Steptoe and Son.

Corbett released a number of 45 rpm records, most of which were novelty songs based upon the rag-and-bone character, including "Harry You Love Her" and "Junk Shop". He recorded a number of sea shanties and folk songs. In 1973, he recorded an album titled Only Authorised Employees To Break Bottles which was a "showcase of accents", with songs from Corbett in a range of accents, including Liverpudlian, Brummie and Mancunian; the title echoes a notice which is visible in the bottle-smashing scene in the film 'The Bargee'. Including the album, he released over 30 songs.

The programme was a success and a full series followed, continuing, with breaks, until 1974, when the Christmas special became the final episode. Although the popularity of Steptoe and Son made Corbett a star, it ended his serious acting as he became irreversibly associated with Harold Steptoe in the public eye. Before the series began, Corbett had played Shakespeare's Richard II to great acclaim; however, when he played Hamlet in 1970, he felt both critics and audiences alike were not taking him seriously and could only see him as Harold Steptoe. Corbett found himself receiving offers only for bawdy comedies or loose parodies of his alter-ego Harold. Production of the sitcom was in the last few years stressful as Brambell was an alcoholic, often ill-prepared for rehearsals and forgetting his lines and movements. A tour of a Steptoe and Son stage show in Australia in 1977 proved a disaster due to Brambell's drinking. During the tour, the pair appeared in character in an advert for Ajax soap powder.

As Prime Minister, Wilson wished to have Corbett appointed an Officer of the Order of the British Empire (OBE), but the middle initial "H" was lost in the process and the award went to the Sooty Puppeteer, Harry Corbett, instead. Both were eventually included in the same New Year's Honours list on 1 January 1976.

The TV episodes were remade for radio, often with the original cast; it is these that were made available on tape and CD. After the Steptoe and Son series officially finished, Corbett played the character again on radio (in a newly written Sketch in 1978), as well as in a television commercial for Kenco coffee. The two men reunited in January 1981 for one final performance as Steptoe and Son in a further commercial for Kenco.

A heavy smoker all his adult life, Corbett had his first heart attack in September 1979. According to his daughter, he smoked sixty cigarettes a day until the heart attack, after which he cut down to twenty. He appeared in pantomime at the Churchill Theatre, Bromley, within two days of leaving hospital. He was then badly hurt in a car accident. He appeared shortly afterwards in the BBC detective series Shoestring, his facial injuries obvious. Other work included a Thames Television/ITV comedy series Grundy and the film Silver Dream Racer with David Essex, both in 1980. In Grundy, Corbett played an old man discovering the permissive society after a lifetime of clean living. The series was a flop and was soon cancelled.

He had a supporting role in the David Essex film Silver Dream Racer in 1980, and also appeared in the controversial film Hardcore in 1977.

Corbett died of a heart attack in March 1982 in Hastings, East Sussex. He was 57 years old. He is buried in the churchyard at Penhurst, East Sussex. The headstone inscription, chosen by his wife Maureen, reads 'The earth can have but earth which is his due: My spirit is thine, the better part of me', from Sonnet 74. Maureen was buried alongside him in 1999. Corbett is commemorated in the name of the Corbett Theatre at the East 15 Acting School at Loughton, which was founded by Margaret Bury and Jean Newlove, two members of Theatre Workshop, where he worked.

The Curse of Steptoe, a BBC TV play about Corbett and Brambell, was broadcast on 19 March 2008 on the British television channel BBC Four, featuring Jason Isaacs as Corbett. The first broadcast gained the channel its highest audience figures to that date, based on overnight returns.