The actress “died peacefully” early on Friday in a London care home Denville Hall which looks after those in the entertainment industry.
Sharing the news, her children noted that Syms “lived an amazing life” and that "she will be so very missed".
Syms is perhaps best known for playing the Queen Mother opposite Helen Mirren as Her Majesty in the 2006 film The Queen or for her role as Sister Diana in the 1958 war film Ice Cold In Alex.
However, the actress also had starring roles in Doctor Who as Mrs Pritchard in the television story Ghost Light and EastEnders as dressmaker Olive Woodhouse.
Her most recent role was in 2019 in the BBC period drama Gentleman Jack as Mrs Rawson.
In a statement, Syms’s children Beatie and Ben Edney said: “Our mother, Sylvia, died peacefully this morning.
“She has lived an amazing life, and gave us joy and laughter right up to the end. Just yesterday we were reminiscing together about all our adventures.
“We would also like to take this opportunity to thank everyone at Denville Hall for the truly excellent care they have taken of our mum over the past year.”
Born on January 6 1934, Syms is also known for the 1961 film Victim, in which she starred with Sir Dirk Bogarde.
Fans have flooded social media have tributes to the late actress.
One user posted: "Sorry to see that we’ve lost the ever-impressive Sylvia Syms (1934-2023), a powerful and astute performer, she was a mainstay of British film and television for seven decades. Her elegant voice was always instantly recognisable. Vale."
A second person shared: "RIP Sylvia Syms, a great presence in British cinema but also someone I once saw fall off the stage of NFT1 during an interview with June Whitfield."
A third wrote: "Beautiful Sylvia Syms - I have loved her since I saw her in Ice Cold in Alex. She was a great treasure - my condolences to @beatieedney on the loss of a very special Mum."