Whittaker will exit the TARDIS in a special episode this autumn, having played the iconic role for the last four years.
Russell T Davies is returning as showrunner for the next series and has selected three-time BAFTA nominee Gatwa, 29, as his new leading man.
"The future is here and it's Ncuti!" Davies said in a statement. "Sometimes talent walks through the door and it's so bright and bold and brilliant, I just stand back in awe and thank my lucky stars.
"Ncuti dazzled us, seized hold of the Doctor and owned those TARDIS keys in seconds. It's an honour to work with him, and a hoot, I can't wait to get started.
"I'm sure you're dying to know more, but we're rationing ourselves for now, with the wonderful Jodie's epic finale yet to come. But I promise you, 2023 will be spectacular!"
Gatwa said: "There aren't quite the words to describe how I'm feeling. A mix of deeply honoured, beyond excited and of course a little bit scared.
"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. I will endeavour my upmost to do the same.
"Russell T Davies is almost as iconic as the Doctor himself and being able to work with him is a dream come true. His writing is dynamic, exciting, incredibly intelligent and fizzing with danger. An actor's metaphorical playground.
"The entire team have been so welcoming and truly give their hearts to the show. And so as much as it's daunting, I'm aware I'm joining a really supportive family.
"Unlike the Doctor, I may only have one heart but I am giving it all to this show."
Filming of Gatwa's first series as The Doctor is expected to get underway imminently for broadcast in late 2023.