Whittaker, 35, best known for her role as Beth Latimer in Bafta-winning drama Broadchurch, described the role as “incredible” and asked fans of the show “not to be scared by my gender”.
She said: “It feels completely overwhelming, as a feminist, as a woman, as an actor, as a human, as someone who wants to continually push themselves and challenge themselves, and not be boxed in by what you’re told you can’t be.
“Doctor Who represents everything that’s exciting about change… the fans have lived through so many changes, and this is only a new, different one, not a fearful one.”
Whittaker revealed she had told “a lot of lies” to keep the announcement secret and used the codename “Clooney” when discussing the role with her husband and agent.
She will team up with the show’s creator Chris Chibnall as he takes over from Steven Moffat as executive producer.
Chibnall said: “After months of lists, conversations, auditions, recalls, and a lot of secret-keeping, we’re excited to welcome Jodie Whittaker as the 13th Doctor.
“I always knew I wanted the 13th Doctor to be a woman, and we’re thrilled to have secured our number one choice.
“Her audition for The Doctor simply blew us all away.”
Capaldi described his successor as a “wonderful actress”.
He said: “Anyone who has seen Jodie Whittaker’s work will know that she is a wonderful actress of great individuality and charm.
“She has above all the huge heart to play this most special part. She’s going to be a fantastic Doctor.”
As well as her role in Broadchurch, Whittaker also featured in romcom One Day, time-travel film Harmony and in both St Trinian’s films.
The Yorkshire-born actress will become the 13th Time Lord.
Peter Capaldi will give up the role after this year’s Christmas special.
Matt Smith took on the role from 2010-2013. He was nominated for a TV Bafta for his portrayal.
David Tennant won Best Actor at the National Television Awards in 2006. He played the Time Lord from 2005-2010.
Christopher Eccleston was the ninth incarnation of Doctor Who and played the character for one series in 2005.
Paul McGann played the eighth Doctor in 1996.
Sylvester McCoy played the time-travelling Doctor from 1987-89.
Colin Baker was the sixth Doctor for three seasons from 1984-86.
The fifth Time Lord was played by Peter Davison from 1982-84.
The fourth incarnation of Dr Who was played by Tom Baker from 1974-81.
Jon Pertwee took on the character between 1970-74. He was the first Doctor to be viewed in colour.
Patrick Troughton played Doctor Who from 1966-69 in black and white.
William Hartnell was the very first Doctor Who, from 1963-66, setting the tone for all future incarnations of the Time Lord.