Claudia – who currently presents a show on the station on Sunday nights – will take over from Graham on Saturday mornings when he steps down at the end of this year.
In a statement, the Strictly Come Dancing host said: “It’s not often I’m speechless but the chance to be with the wonderful Radio 2 listeners every Saturday has left me, quite frankly, gobsmacked.
“I hope my voice comes back in time for the first show as I can no longer simply rely on a fake tan and a fringe. There’s nobody I’d rather be with at the weekend, it’s a privilege and an honour.”
Commenting on the news, Graham tweeted: ”Congratulations @ClaudiaWinkle You’ll love it!! Welcome to @BBCRadio2 Saturday mornings!!"
After 10 years with Radio 2, Graham recently announced he would be hosting his final show with the broadcaster next month.
It was later confirmed that he’d signed a new deal with Virgin Radio UK, where he will be presenting a new show on Saturdays and Sundays in 2021.
He said: “I am excited and a little surprised to be joining Virgin Radio UK in 2021.
“I was very content where I was but the opportunity to host shows across the weekend seemed too good to miss out on. Plus the energy and enthusiasm at Virgin Radio are infectious and I can’t wait to get started!”
He then joked: “Did I mention the studios are very close to my house?”
After announcing he was leaving Radio 2, Graham insisted that he would continue working with the BBC on his usual talk show and the UK’s Eurovision coverage, as well as judging the new series of RuPaul’s Drag Race UK.
“Happily with the chat show, Eurovision and Drag Race, the BBC continues to be my perfect TV home,” he said at the time.