The Olivier Award-winning actor and director was diagnosed with terminal cancer earlier this year, and his death was announced by the Royal Shakespeare Company (RSC) on Friday.
Dame Judi, who starred alongside Sir Antony in the 1997 film Mrs Brown, was among those paying tribute.
The 86-year-old described his performance as former prime minister Benjamin Disraeli as “spectacular”.
Speaking to BBC Radio 4’s PM programme, she said: “He could completely immerse himself in a character and make it completely remarkable, but not necessarily on his own terms.
“He was sublime. He was totally engrossed whenever he was working in that part and in that character.
“He was one of those remarkable actors who reserved that incredible intensity for the time he was on the stage.
“And he could be very relaxed and part of a company off it, but was very much his own person.
“Very retained, very principled, very single-minded about the decisions he made. Which in a way is what made him so remarkable.”
Brian Blessed, who performed alongside Sir Antony in Richard III in Stratford-upon-Avon, told the programme: “He revolutionised Richard III entirely. Amazing imagination, amazing vocal power. He hobbled around the set like a great bottled spider. He would terrify the audience in the first few rows.”
Blessed said to be on stage with Sir Antony was “mind-blowing” and added: “It was from another century. It was from another galaxy.”
The National Theatre posted a statement on Twitter from director Rufus Norris, saying: “With the tragic passing of Antony Sher, one of the great titans has left us.
“His contribution and example to our theatre world was exemplary, and his standing within the ranks of National Theatre actors could not be higher.”
Sir Antony’s husband, Gregory Doran, the RSC’s artistic director, announced in September that he was taking a period of compassionate leave to care for Sir Antony.
The South African-born actor tied the knot with Doran on December 21 2005, the first day same sex couples could legally form a civil partnership in the UK.
Sir Antony starred in a number of RSC productions, including a role in 2016 in King Lear, as well as playing Falstaff in the Henry IV plays and Willy Loman in Arthur Miller’s Death Of A Salesman.
He was the Prince of Wales’ favourite actor – a fact the royal revealed during his 2017 Commonwealth Tour.
Earlier landmark performances included Leontes in The Winter’s Tale, Iago in Othello, Prospero in The Tempest and the title roles in Macbeth and Tamburlaine The Great, as well as his career-defining Richard III.
He moved to Britain to study drama in the late 1960s and joined the RSC in 1982. His breakthrough role came two years later in Richard III, a part which earned him the best actor accolade at the Olivier Theatre Awards.
His theatrical skills were not limited to the West End, and his adaptation of If This Is A Man, by Primo Levi, into a one-man show titled Primo, ran on Broadway.
Off stage he had roles in films including Shakespeare In Love and Mrs Brown, and played Adolf Hitler in 2004’s Churchill: The Hollywood Years.
His final production with the RSC was Kani’s Kunene And The King, which saw him star opposite Kani as Jack, an actor acclaimed for his roles in Shakespeare who is diagnosed with liver cancer.
RSC executive director Catherine Mallyon and acting artistic director Erica Whyman said in a statement: “We are deeply saddened by this news, and our thoughts and sincere condolences are with Greg, and with Antony’s family and their friends at this devastating time.
“Antony had a long association with the RSC and a hugely celebrated career on stage and screen.
“Antony’s last production with the company was in the two-hander Kunene And The King, written by his friend and fellow South African actor, writer and activist, John Kani.”
The statement added: “Antony was deeply loved and hugely admired by so many colleagues.
“He was a ground-breaking role model for many young actors, and it is impossible to comprehend that he is no longer with us.
“We will ensure friends far and wide have the chance to share tributes and memories in the days to come.”
The RSC said Doran will remain on compassionate leave and is expected to return to work in 2022.