She played the hilariously snobbish character on the BBC program in the early 1990s, but also had a prolific career in the theatre.
Her agent said she died peacefully in her sleep on Friday "surrounded by love".
"Even at 96 years old, Dame Patricia's passion for her work and for connecting with live audiences never waned, just as new generations of audiences have continued to find her through her beloved television roles," a statement to the PA news agency said.
"She will be dearly missed by those closest to her and by her devoted admirers around the world."
Written by Roy Clarke, Keeping Up Appearances also starred Clive Swift as Hyacinth's long-suffering husband Richard in the series, which ran from 1990 to 1995 and also became popular in the US.
The BBC's director of comedy, Jon Petrie, said in a tribute that Routledge was an actor of "remarkable range", adding: "But her portrayal of Hyacinth Bucket in Keeping Up Appearances has to be one of the most iconic performances in British comedy.
"She took a character on the page and gave her such truth, precision and warmth that Hyacinth became part of the national conversation; instantly recognisable, endlessly quotable, and loved around the world."
Routledge won an Olivier Award for her role as the Old Lady in Leonard Bernstein's operetta Candide in 1988, a Tony Award for her part as Alice Challice in Darling of the Day in 1968, and worked in a number of productions across six decades.
She was born in February 1929 in Birkenhead, Merseyside, where her father was a high-class gents' outfitter and "keeping up appearances" was the name of the game.
She read English at Liverpool University where she appeared in student shows, later making her professional debut at the Liverpool Playhouse in 1952 as Hippolyta in A Midsummer Night's Dream.
She made her acting debut in London two years later and went on to became well known on Broadway, appearing in musicals and the play How's The World Treating You?
Alongside the stage she also made her mark on TV and became known for playing Kitty in Victoria Wood: As Seen on TV in the mid-1980s before playing Hyacinth in the BBC's Keeping Up Appearances.
In 2017 she was honoured at Buckingham Palace as she was made a Dame Commander of the British Empire for her services to the theatre and charity.
She starred in BBC crime drama series Hetty Wainthropp Investigates, where she donned the guise of a woman who decides life begins at 60 as she becomes a private eye.
Routledge also starred in Alan Bennett's Talking Heads TV monologues, including A Woman of No Importance and A Lady of Letters.
One of Routledge's notable film roles saw her star opposite Sidney Poitier in 1967's To Sir, With Love.
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