Smith was one of a select few to win the treble of an Oscar, Emmy and Tony during seven decades on stage and screen, becoming a star known for her sharp intelligence and waspish wit.
United Kingdom Prime Minister Keir Starmer said Smith "introduced us to new worlds with the countless stories she acted over her long career".
"She was beloved by so many for her great talent, becoming a true national treasure whose work will be cherished for generations to come," he said.
After starting on stage in the 1950s, Smith became a fixture at the UK's new National Theatre in the 1960s, working alongside Laurence Olivier, before winning her first Oscar at the end of the decade.
But for many younger fans in the 21st century, she was best-known as Professor McGonagall in all seven Harry Potter movies, and the Dowager Countess in the hit TV series Downton Abbey, a role that seemed tailor-made for an actress known for purse-lipped asides and malicious cracks.
She died in hospital in London early on Friday, her sons Chris Larkin and Toby Stephens said.
"An intensely private person, she was with friends and family at the end," they said in a statement.
Smith's first Academy Award nomination was for her turn playing Desdemona opposite Laurence Olivier's Othello in 1965, before winning the Oscar for her role as an Edinburgh schoolmistress in 1969's The Prime of Miss Jean Brodie.
She won her second Oscar for her supporting role in the 1978 comedy California Suite.
Other critically acclaimed roles included Lady Bracknell in Oscar Wilde's The Importance of Being Earnest on the West End stage, a 92-year-old bitterly fighting senility in Edward Albee's play Three Tall Women and her part in 2001 black comedy movie Gosford Park.
In 1990 Smith was knighted by Queen Elizabeth and became a Dame.
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