Dimbleby will, however, continue to chair sister show Any Questions? and the two programmes will "remain closely linked in the issues they address", the BBC said.
Anand brings a wealth of experience to Any Answers?, after working for over a decade on BBC Two's Daily Politics programme with Andrew Neil, along with stints on Radio 5 Live and BBC World Service.
She has also previously presented on various Radio 4 programmes, including Beyond Westminster, Westminster Hour, Saturday Live and Midweek.
Radio 4 controller Gwyneth Williams said that she was "delighted" to welcome Anand as the new presenter of Any Answers?.
"She knows her politics and is a tried and tested broadcaster who is interested in the lives and experiences of our listeners," she said.
"Any Answers? is a cornerstone in the Radio 4 schedule - I can't think of anyone better to work with Jonathan, who will continue to chair Any Questions? every week in his usual brilliant way."
Any Answers? was first established in 1954 to give listeners the chance to respond to issues raised in its sister programme Any Questions?. The show switched to a live phone-in programme in 1989.
Dimbleby become chair of Any Questions? in 1987 and started presented Any Answers? the following year, but he has now taken the "hard decision" to hand on the baton for the latter show.
"It has been a privilege and pleasure to share so many conversations with listeners over these years. I shall miss that greatly," he said.
"But, as I am not a politician, I hope I will not be misinterpreted if I say that I am handing over to Anita simply because I want to spend more time with my family, which means less time at the microphone."
Anand said that it was an "honour" to be asked to present the Radio 4 show, adding that "getting to the heart of the stories that really affect people is why I became a journalist".
"I'm excited by the opportunities afforded by the programme to engage with listeners throughout the country and to hear their direct experiences of the big policy and political issues debated on Any Questions?," she said.
"I have admired Jonathan Dimbleby for many years and I take the stewardship of this wonderful programme very seriously indeed."
Anand will present her first edition of Any Answers? on June 9 at 2pm.