Sir Peter, who won an Academy Award for Amadeus and whose West End and Broadway award-winning play Equus starred the 26-year-old actor, passed away in Marymount Hospice in Cork, Ireland, on Mondy after a short illness.
Radcliffe said he was one of the "greatest playwrights" ever.
The Harry Potter star wrote: "Peter Shaffer was and remains one of the UK's greatest playwrights. He created challenging, moving and fantastically theatrical work and was also an incredibly kind, generous and funny man."
In 2007 and 2008, at the height of his Harry Potter fame, Radcliffe starred in the London and New York productions of Shaffer's 1973 play in the lead role of Alan Strang.
It followed the story of a disturbed teenager (Strang) who blinds six horses. The actor admitted he would be "forever grateful" for the opportunity to play the intricate character.
Radcliffe was 17 when he starred in the play. "In my career Equus remains one of the most important and valuable things I've ever done in terms of how much I learnt from it.
"I will forever be grateful to him for trusting me to play the character of Alan Strang at a time when I had extremely limited experience on stage. I feel very privileged to have worked with him and will miss him."
Sir Peter wrote more than 18 plays and several have been turned into films, including Equus and Amadeus, which was published in 1979, and adapted for the big screen in 1984 starring F. Murray Abraham and Tom Hulce.
It won eight Academy Awards including the Best Adapted Screenplay Oscar as well as the Golden Globe Best Screenplay the following year.
Much of his work was put on at The National Theatre and Rufus Norris, director of the National Theatre, also paid tribute on the company's site.
"Peter Shaffer was one of the great writers of his generation and the National Theatre was enormously lucky to have had such a fruitful and creative relationship with him," Norris said.
"The plays he leaves behind are an enduring legacy."
