In a statement, he said that “if Mum had the first idea I might not have shown up today because of her she would have been furious”.
One listener said the announcement should have come with a “mascara warning”.
In the message, read by sports reporter Vassos Alexander, Evans told his millions of Radio 2 listeners: “The reason I’m not with you today is because just before we came on air my mum passed away and I needed to go straight back home to be with the family.
“But it’s all OK. In fact, it’s very OK. Mum needed to be at peace.
“Moreover, she deserved to be at peace. We’ve been ready for this for some time now and, as she has throughout her life, in death she has brought us together.”
The 52-year-old former Big Breakfast host added: “She was an incredible woman.
“Anyone who has ever met her will tell you that, and ultimately there was no battle lost, only a life won. Every single day.”
Alexander said reading Evans’ words was “difficult for me because we’re talking about one of my best friends”.
Listeners said they had been moved by the statement.
JoeyAC tweeted: “What an emotional message. Such lovely words.”
Paula Ferguson, who said she had been “in tears”, added that the announcement should have come with a “mascara warning”.
The “wonderful message” had Grocer On The Green “in total tears”.
And the Archbishop of York, John Sentamu, tweeted: “My dear friend, I’m so sorry to hear of the death of your beloved mother Minnie!
“Your statement read out on your breakfast show was Brill: A life lived not lost! Yah Man! Death isn’t the end! The end is life in God! God Bless: WITH YOU!”
Evans, who was said to be very close to his mother, tweeted a picture in January last year of her eating what he said was her “first ever slice of pizza”.
Recently, he announced that his wife Natasha was expecting twins.
The star wished his listeners a “lovely Thursday”, adding: “See you tomorrow.”