Twitter user Jenny Trout posted the video of her daughter watching the announcement, which was made after the Wimbledon finals.
It showed her daughter watching the TV with a slight frown as she waited for the news, before her face lit up and she shrieked "The new Doctor is a girl!" to the camera.
The tweet has garnered more than 17,000 retweets and 60,000 likes.
Whittaker will replace Peter Capaldi on the show and become the 13th actor to play the iconic role.
Other actors to have played the Doctor in recent years include David Tennant, Matt Smith and Christopher Eccleston. However a woman had never held the role before, despite Time Lords being proven to be able to regenerate as women.
Whittaker said: "I'm beyond excited to begin this epic journey - with Chris [Chibnall, showrunner] 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."
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Chibnall 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."
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