Eighteen-year-old Gibson was previously on Coronation Street.
The BBC said Doctor Who will return in November next year with three special episodes to coincide with the 60th anniversary, with David Tennant as the 14th Doctor.
Gatwa, 30, will then take control of the Tardis, with his first episode airing during the festive period in 2023.
He added: “This role and show means so much to so many around the world, including myself, and each one of my incredibly talented predecessors has handled that unique responsibility and privilege with the utmost care.”
It has already been announced that writer Russell T Davies will return as the programme’s showrunner after departing in 2009.
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Davies, who was responsible for Doctor Who’s revival in 2005, will be back to celebrate the show’s 60th anniversary in 2023.
“Ncuti dazzled us, seized hold of the Doctor and owned those Tardis keys in seconds,” he said earlier this year.
“It’s an honour to work with him, and a hoot, I can’t wait to get started.”