Announcing her death, her agent's statement added: "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.
"She will be dearly missed by those closest to her and by her devoted admirers around the world."
The Birkenhead-born star was celebrated for her performances on stage as well as screen after starting her career in the theatre in the 1950s.
Following roles in London's West End, she established herself in the US when she won the Tony Award for best lead actress in a musical on Broadway in New York for Darling of the Day in 1968.
She was as adept at serious drama as she was at musicals and comedy - she appeared in Richard III and Henry V for the Royal Shakepseare Company, then won an Olivier Award for best actress in a musical for Candide in the 1980s.
The 80s also saw her become a true national TV star - playing Kitty in Victoria Wood's As Seen on TV, and delivering three of Bennett's renowned Talking Heads monologues, including 1988's A Lady of Letters, for which she received a Bafta Award nomination.
The actress appeared numerous times at Chichester Festival Theatre near her home in West Sussex, and the venue's artistic director Justin Audibert and executive director Kathy Bourne paid tribute to her as "irreplaceable, inimitable and unforgettable".
"Patricia Routledge was a supreme actress. Her talents were particularly associated with comedy, though her range was far greater than that," they said in a statement.
"She was not only a national treasure, she was a Chichester treasure too - a stalwart supporter of the city and community, and of the Festival Theatre whose stage she graced for five decades."
She was made an OBE in 1993, a CBE in 2004, and a dame in 2017.