Elliott was born in London, the son of Nina (née Mitchell) and Myles Laymen Farr Elliott. Myles was a barrister who had read law and Arabic at Cambridge before fighting with the Glosters at Gallipoli and in Mesopotamia. In 1930, Myles Elliott was appointed solicitor-general to the Mandatory Government in Palestine. Three years later, following a series of controversial government prosecutions, he was assassinated outside the King David Hotel, and buried in the Protestant Cemetery on Mount Zion. Elliott's elder brother Neil was land agent to Lady Anne Cavendish-Bentinck.

Privately bisexual, Elliott was married twice: first to Actress Virginia McKenna for a few months in 1954, and later, in an open marriage, to American Actress Susan Robinson (7 March 1942 – 12 April 2007), with whom he had two children, Mark and Jennifer. Jennifer committed suicide by hanging in 2003.

After making his film debut in Dear Mr. Prohack (1949), he went on to play a wide range of parts, often ineffectual and occasionally seedy characters, such as the drunken Journalist Bayliss in Defence of the Realm, the Criminal abortionist in Alfie, and the washed-up film Director in The Apprenticeship of Duddy Kravitz. Elliott and Natasha Parry played the main roles in the 1955 television play, The Apollo of Bellac. He took over for an ill Michael Aldridge for one season of The Man in Room 17 (1966)

Elliott made many television appearances, which included plays by Dennis Potter such as Follow the Yellow Brick Road (1972), Brimstone and Treacle (1976) and Blade on the Feather (1980). He starred in the BBC's adaptation of Charles Dickens's short story The Signalman (1976).

In the 1980s, he won three consecutive British Academy of Film and Television Arts (BAFTA) awards – Best Supporting Actor for Trading Places as Dan Aykroyd's kindly butler, A Private Function, and Defence of the Realm, as well as an Academy Award nomination for A Room with a View. He also became familiar to a wider audience as the well-meaning but addlepated Dr. Marcus Brody in Raiders of the Lost Ark (1981) and Indiana Jones and the Last Crusade. A photograph of his character appears in Indiana Jones and the Kingdom of the Crystal Skull, and a reference is made to Brody's death. Also, a statue was dedicated to Marcus outside Marshall College, the school where Indy teaches. In 1988, Elliott was the Russian mole Povin, around whom the entire plot revolves, in the television miniseries Codename: Kyril.

Having filmed Michael Winner's The Wicked Lady (1983), Elliott was quoted in a BBC Radio interview as saying that Marc Sinden and he "are the only two British actors I am aware of who have ever worked with Winner more than once and it certainly wasn't for love. But curiously, I never, ever saw any of the same crew twice." (Elliott in You Must Be Joking! (1965) and The Wicked Lady and Sinden in The Wicked Lady and Decadence). Elliott had worked with Sinden's father, Sir Donald Sinden, in the film The Cruel Sea (1953). He co-starred with Katharine Hepburn and Harold Gould in the television film, Mrs. Delafield Wants to Marry (1986) and with Nicole Kidman in Bangkok Hilton (1989).

In 1988, Elliott was appointed a Commander of the Order of the British Empire (CBE) for his services to acting. His career included many stage performances, including with the Royal Shakespeare Company, and a well-acclaimed turn as the twin brothers in Jean Anouilh's Ring Round the Moon. His scene-stealing abilities led Gabriel Byrne, his co-star in Defence of the Realm, to say: "Never act with children, dogs, or Denholm Elliott."

Elliott was diagnosed with HIV in 1987 and died of AIDS-related tuberculosis at his home in Santa Eulària des Riu on Ibiza, Spain, on 6 October 1992 at the age of 70. Tributes were paid by actors Sir Donald Sinden and Sir Peter Ustinov, Playwright Dennis Potter, and former wife Virginia McKenna. Sinden said, "He was one of the finest screen actors and a very special actor at that. He was one of the last stars who was a real gentleman. It is a very sad loss." Ustinov said, "He was a wonderful actor and a very good friend on the occasions that life brought us together." Potter commented, "He was a complicated, sensitive and slightly disturbing actor. Not only was he a very accomplished actor, he was a dry, witty and slightly menacing individual. As a man, I always found him very open, very straightforward and very much to the point." McKenna added: "It is absolutely dreadful, but the person I am thinking of at the moment more than anybody is his wife. It must be terrible for her." Ismail Merchant described Elliott as "an all-giving person, full of life ... He had an affection and feeling for other actors, which is very unusual in our Business."

His widow set up a charity, the Denholm Elliott Project, and collaborated on his biography. She worked closely with the UK Coalition of People Living with HIV and AIDS. Susan Robinson Elliott died on 12 April 2007, aged 65, in a fire in her flat in London.