The new Doctor’s identity was revealed in a trailer broadcast at the end of the Wimbledon men’s singles final.
The Broadchurch star succeeds Peter Capaldi, who took over the role in 2013 and leaves in the forthcoming Christmas special.
Whittaker, 35, said it was “overwhelming, as a feminist” to become the next Doctor.
She will make her debut on the sci-fi show when the Doctor regenerates in the Christmas Day special.
The Huddersfield-born star, who was a late favourite to become the Doctor, will find a familiar face for her on set – Doctor Who’s new showrunner is Broadchurch creator Chris Chibnall.
Whittaker said: “I’m beyond excited to begin this epic journey – with Chris and with every Whovian on this planet.
“It’s more than an honour to play the Doctor. It means remembering everyone I used to be, while stepping forward to embrace everything the Doctor stands for: hope. I can’t wait.”
The actress also shares another Broadchurch link with Doctor Who – co-star David Tennant was the 10th Doctor.
Whittaker said it felt “incredible” to take on the role, saying: “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 and can’t be.”
And she told fans not to be “scared” by her gender.
“Because this is a really exciting time, and Doctor Who represents everything that’s exciting about change,” she said, adding: “The fans have lived through so many changes, and this is only a new, different one, not a fearful one.”
Whittaker said she had used the codename “Clooney” when discussing the part with her husband and agent – as actor George is “an iconic guy”.
Chibnall said the 13th Doctor was always going to be a woman.
He said: “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. Jodie is an in-demand, funny, inspiring, super-smart force of nature and will bring loads of wit, strength and warmth to the role. The 13th Doctor is on her way.”
Chibnall is taking over from Steven Moffat, who leaves the series at the same time as Capaldi.
Capaldi, who had said he wanted to see a woman replace him, 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.”
Dedicated Whovians were quick to react to the news of Jodie Whittaker taking over the Tardis.
On social media, some said it would encourage them to watch the show for the first time – but others said the casting meant they would be switching off, and that the Doctor should be played by a man.
Carla Joanna tweeted to say that she would be tuning in and that the trailer “made me choke up a little”. Another tweeter, Ayad, said: “I don’t even watch Doctor Who but a woman doctor is so cool.”
But Samantha Melton said: “I am a woman and a feminist but I don’t want a female Doctor. To me it’s trying too hard to tick the boxes.”
Doctor Who writer Jenny Colgan, who has written for the series’ books and audio dramas, said: “I am of course incredibly excited the new Doctor is a woman; Steven Moffat has been paving the way for this for ages and it is absolutely about time.
“I can’t imagine what it’s like for Jodie: she must be so scared and excited all at once, but I couldn’t be happier, and 100% can’t wait to write for her.”
Will Howells, who writes for the Doctor Who magazine and has been a fan for 25 years, said: “In 2017, there shouldn’t be anything major about a TV series changing from a male lead to a female one. We’ll also maybe see a solo male companion as a regular feature for the first time.
“I don’t think it’s a risky choice at all – but if a show that can go anywhere and do anything can’t take risks, what can?”
And science fiction and fantasy author Paul Cornell said: “It’s always been time for a woman Doctor and it’s great we got there.
“Well done to Steven Moffat for laying the groundwork. She’s going to be amazing. And that first episode of hers is going to get a lot of new people watching.”
Whittaker starred as Beth Latimer in the three series of the ITV crime drama Broadchurch, as the mother of a murdered boy.
As well as TV work, Whittaker has appeared on the big screen, in One Day, Attack the Block and St Trinian’s. She made her film debut in 2006’s Venus, opposite Peter O’Toole.
Traditionally, each Doctor has their own distinctive look, raising questions about the cloak Whittaker wears in the trailer. However, she has said it is not part of her official Doctor Who outfit, and that she does not yet know what she will wear.