Announcing the news on this morning's show, Wogan said that he was sure his listeners would welcome Evans with "loyalty and support".
Wogan said: "It touches me deeply that I've played a part in your lives for, it seems, like generations.
"This is not goodbye - not even au revoir. As we used to say when I was a boy, see you later alligator."
Wogan first hosted the breakfast show in 1972 and returned as its presenter in 1993, while Evans currently fronts Radio 2's drivetime programme.
Last month, commercial station trade body RadioCentre claimed that there would be a risk of the channel losing its older listeners if Wogan were replaced by Evans.