Born in Richmond, Surrey, and nicknamed "Betty", Johnson was the second daughter of Robert and Ethel (née Griffiths) Johnson. Her first public performance was in 1916, when she played a role in a charity performance of King Cophetua and the Beggar Maid to raise funds for returned First World War Soldiers.

She attended St Paul's Girls' School in London from 1919 until 1926, and played in the school's orchestra under Gustav Holst. She acted in school productions, but had no other acting experience, when she was accepted to study at Royal Academy of Dramatic Art in 1926, where she was in the same class as Margaretta Scott. She later spent a term in Paris, studying under Pierre Fresnay at the Comédie Française. She later recalled her choice of an acting career with the comment, "I thought I'd rather like it. It was the only thing I was good at. And I thought it might be rather wicked."

Her stage début, and first professional role, was as Sarah in George Bernard Shaw's Major Barbara at the Theatre Royal, Huddersfield in 1928. She went to London the following year to take the place of Angela Baddeley in the part of Currita in A Hundred Years Old, which was performed at the Lyric Theatre in Hammersmith. In 1930 Johnson played in Cynara with Sir Gerald Du Maurier and Dame Gladys Cooper. She made her first trip to the United States the following year to star as Ophelia in a New York City production of Hamlet.

She returned to London, where she appeared in a number of minor productions, before establishing herself with a two-year run in The Wind and the Rain (1933–35). She married the Journalist Peter Fleming in 1935, and in 1939 gave birth to their first child, a son. Her theatre career flourished with her portrayals of Elizabeth Bennet in Pride and Prejudice (1940) and the second Mrs. de Winter in Rebecca (1940); the production of the latter was halted when the theatre was destroyed by a Luftwaffe bomb in September 1940.

Johnson was married to Peter Fleming from 1935 until Fleming's death from a heart attack in 1971, while on a shooting expedition near Glencoe in Argyll, Scotland. Fleming was the brother of the James Bond creator Ian Fleming.

During the Second World War, Johnson lived with her widowed sister and sister-in-law and helped care for their combined seven children. Unable to commit her time to the often lengthy run of a play, Johnson preferred the less time-consuming schedules of film and radio, that allowed her to devote time to her family, and her work for the Women's Auxiliary Police Corps. She appeared in In Which We Serve (1942) and This Happy Breed (1944), both directed by David Lean and written by Noël Coward.

Lean and Coward sought Johnson for the next production, Brief Encounter (1945). She accepted the role with misgivings because of her family responsibilities, but was interested in the part, writing to her husband, "There is no getting away from the fact that it is a very good part and one which I should love to play. I have found myself already planning how I should play bits and how I should say lines..." A romantic drama about a conventional middle class housewife who falls in love with a Doctor she meets in the refreshment room at a railway station, the film was well-received, and is now regarded as a classic. Johnson was awarded the New York Film Critics Circle Award for Best Actress and was nominated for the Academy Award for Best Actress.

After the war, Johnson concentrated on her family life, which included two daughters born in 1946 and 1947 and her occasional acting work was secondary for the following decade.

In 1952, she opened The Grass is Greener. In 1957 she acted with Ralph Richardson in The Flowering Cherry. As a member of Laurence Olivier's National Theatre Company, Johnson appeared in the plays The Master Builder (1964) with Olivier and Hay Fever (1965), and later reprised her roles in the television productions.

For her role in The Prime of Miss Jean Brodie (1969), she received the BAFTA Award for Best Actress in a Supporting Role. She was created a Commander of the Order of the British Empire (CBE) in 1958, "for services to the theatre", and was raised to Dame Commander (DBE) in 1981.

In 1982, she was touring with Sir Ralph Richardson in Angela Huth's The Understanding and the play's West End run had been announced. On one of her days off, she was at her home in Nettlebed, Oxfordshire playing bridge with friends, when she collapsed from a stroke. She died a few hours later in her home. She left an estate worth £150,557.

Since the late 1990s, the two sisters, Kate Grimond and Lucy Fleming, have co-owned the Ian Fleming estate.

On 18 December 2008, to mark the centenary of her birth, a blue plaque was unveiled at her childhood home in Richmond. Among the guests at the ceremony were her daughters, Lucy Fleming and Kate Grimond. In The Times, Grimond noted that the "tragedy of theatre" is that even the best performances fade from memory, and that her mother's current reputation rests almost entirely on her performance in Brief Encounter. Grimond noted that the advent of video allowed the film to be seen by a new audience, and that modern appraisals of the film had led to its being regarded as a classic.