William Roache, who has played Ken Barlow since the first episode, said he would be “so desperately missed”.
Coronation Street executive producer Kieran Roberts called Warren “a pioneer, a revolutionary, a true genius [and] a giant of British television”.
Warren, who started the Granada Television show when he was 24, died on Tuesday after a short illness.
Born in Lancashire in 1936, he wrote episodes for the ITV soap until the late 1970s.
Roache described the writer, who continued to visit the soap’s set in Trafford until recently, as the “father” of the soap.
“When I first met Tony, I couldn’t quite believe he’d created and written Coronation Street, because he was no more than a young boy,” he said.
He added the writer had a “boyish energy” that never left him: “I loved Tony’s energy. He was the father of Coronation Street and he gave us all so much.”
Helen Worth, who worked alongside Warren for 42 years as the character Gail McIntyre, said Warren was “a genius of our time [and] the dearest, funniest and most inspirational man of his generation”.
“He brought real life into our homes for us all to relate to and enjoy. He will, of course, live on forever through Coronation Street,” she said.
Coronation Street actors Kym Marsh, Anthony Cotton and Samia Ghadie were among those who paid tribute to Warren on Twitter.
Cotton said the writer left “the greatest legacy”, while Ghadie said he had been “a truly wonderful man” and Marsh described him as “amazing”.
Warren was born Anthony McVay Simpson in Eccles, Salford, in 1937, and took the stage name of Warren during his career as a child star.
He trained at Liverpool’s Elliott Clarke Theatre School and was a regular on the BBC radio show Children’s Hour, before acting in radio plays alongside some of the actors who would become household names because of Coronation Street, including Violet Carson and Doris Speed.
His idea for Coronation Street was commissioned for 13 episodes by Granada in 1960 and the show has gone on to be one of the UK’s most successful ever, reaching viewing figures of 26.6 million for the departure of the much-loved Hilda Ogden in 1987.
The writer was made an MBE in 1994 for his services to television drama and remained a consultant on the soap until his death, with his creator credit appearing at the start of the closing credits of every episode.
A drama based on his life and the creation of the show, The Road to Coronation Street, was broadcast on BBC Four in 2010.