The 47-year-old presenter said that being the first woman to host the UK's most listened to show would be "both an honour and a privilege".
Ball, who also hosts Strictly Come Dancing: It Takes Two, said: "I'm absolutely thrilled to be following in the giant footsteps of Chris Evans as the host of the Radio 2 Breakfast Show.
"To be the first woman to present this very special show is both an honour and privilege.
"Believe me, I'm not under-estimating the enormity of the task ahead - to follow not one but two of my broadcasting idols into such a well-loved show is somewhat daunting.
"But I hope, in the same way that Chris made this show his own after taking over from the wonderful Sir Terry Wogan, that, with a top team alongside me, I can bring the fabulous Radio 2 audience a show they want to wake up to."
The move had been teased by Evans earlier in the morning, with the 52-year-old telling listeners after playing a trail for a Fatboy Slim documentary: "This family is taking over, more of that later."
Ball was married to the DJ, real name Norman Cook, for 18 years before they announced their separation in 2016.
It is not the first time she has replaced Evans in a role, having presented Channel 4's The Big Breakfast in the 1990s following his departure from the show.
She has also made history as a first BBC female solo breakfast show host before, after helming the Radio 1 breakfast on her own from September 1998 to March 2000.
Bob Shennan, director of BBC radio and music, said: "The Radio 2 Breakfast Show is arguably the most coveted job in UK radio with a long and illustrious history, and I'm thrilled that the baton is now passing to Zoe, who I know will make the show her very own."