Gatwa, who is best known for his portrayal of Eric in Netflix's Sex Education, is set to replace Jodie Whittaker in the role later this year with Russell T Davies also returning as showrunner.
Gillan, who played Amy Pond between 2010 and 2013, excitedly tweeted: "We have a new doctor!!!!!"
Sylvester McCoy, who played the seventh Doctor tweeted his approval saying: "Welcome Ncuti, and especially delighted to welcome another Scot!"
Former Doctor Matt Smith, talking at the Paris Fan Festival, said: "I think he's going to be brilliant, I think he's going to be truly brilliant. I think it's an inspired piece of casting, and along with Russell, who's such a great leader of that show."
Georgia Tennant, who is married to tenth doctor David Tennant and also played the Doctor's daughter in an episode of the show, joked by saying: "Hello dad."
There were also reactions from various Sex Education cast members including Emma Mackey who posted the news on her Instagram and wrote "my doctor" accompanied with love heart emojis.
Asa Butterfield also tweeted: "Incredible, the most deserving man out there. Wowowowowow!!"
Alistair Petrie also posted his enthusiastic reaction to the news: "Fearsome talent, he’ll be glorious. Thrilled and very proud. Go The Gatwa."
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Ncuti also discussed winning the role, saying in a statement: "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."
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."
Davies also commented on the casting: "The future is here and it’s Ncuti! 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."
The It's a Sin creator also urged fans to watch the conclusion of Whittaker's tenure: "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."
Watch below: Sex Education star Ncuti Gatwa 'honoured' to take over from Jodie Whittaker