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".
Tributes were also paid to Warren from actors on other soaps, including from EastEnders' Jessie Wallace and Adam Woodyatt, and Emmerdale's Gemma Oaten.
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.
Daran Little, who wrote for Coronation Street before penning the show about Warren, said on Twitter that "Tony was my dad in TV".
"Without his encouragement, I would never had a career," he said.
In 2007, Warren spoke to the Manchester Evening News about the prejudice he faced as a gay man writing the soap before decriminalisation in 1967.
He said while "a lot of creative people at Granada didn't care", he faced a lot of homophobic remarks from some staff.
Describing how he confronted the abuse with the statement "you call my brothers, you call me", he said he "didn't know I felt so strongly until that moment, and from then on I never pretended to another soul that I was anything other than what I am".
Receiving an award in Salford in November last year, Warren joked that he had been "haunted" by his Coronation Street characters for 50 years.
He said the one that haunted him the most was Ena Sharples, adding he was "left wondering just what Ena would have made of this [award]".
"She'd say, 'Oh yes, yes, very nice, very nice. One more ornament to dust. Not much dusting done in your house anyway.'
"It puts me in mind of one thing and one thing only. I bet it's not occurred to you but it's occurred to me. This could be the last monument you see before the actual gravestone."
An ITV spokesman said the "legendary creator and acclaimed writer" died on Tuesday night "surrounded by his loving friends after a short illness".
Tributes have been paid to Tony Warren, the Coronation Street creator and writer who has died aged 79.