The actor and Time Team presenter joined Blackadder producer John Lloyd in hailing Byrne, who became well-known for playing Nursie in the second series of the hit 1980s comedy.
Denville Hall, a care home for elderly actors in north-west London, said she died on June 17.
Sir Tony told BBC Five Live she had been like a mother to "her boys" on the show and was "indulgent" towards them despite being an experienced actress whose past included playing roles for the Royal Shakespeare Company.
He said: "She had unerring skill to make the daftest, silliest thing sound absolutely real and absolutely plausible."
Byrne was born in Ashford, Kent and studied at drama school before joining the RSC. She also starred alongside Sir Tony in an episode of the children's sitcom Maid Marian And Her Merry Men.
Byrne is the second star of Blackadder to die in the last fortnight. Rik Mayall, who played Lord Flashheart in the second and fourth series died at his home in south-west London, last week.
The two shared screen time during the second series of Blackadder in 1986.