The two-part play - a follow-up to J.K. Rowling's "Harry Potter" series of books - won in nine of the 11 categories it was nominated in at the ceremony at London's Royal Albert Hall.
"Cursed Child" won Best New Play, Best Actor (Jamie Parker), Best Actor in a Supporting Role (Anthony Boyle), Best Actress in a Supporting Role (Noma Dumezweni) and Best Director (John Tiffany).
It also won in the Best Lighting Design, Best Costume Design, Best Sound Design and Best Set Design categories.
"People sometimes ask me how I do it, and here's the answer - we do it. I was inspired to dream and my advice to everyone is: dream big. There are no mistakes only lessons.
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"We can help to change the world just a little bit. I've seen the evidence. Shakespeare was right: we are such stuff that dreams are made of. Keep making stuff keep dreaming," Branagh said.
Actress Judi Dench praised the actor-director saying, "I am so glad you are getting this award. And please employ me again'', while Mark Rylance introduced the actor and director with the words: ''He makes it seem as easy. I know he has grave doubts, full of fears like any of us, and yet he turns up."
"Groundhog Day" won Best New Musical and Best Actor in a Musical for Andy Karl, while "Glee" actress Amber Riley won Best Actress in a Musical for "Dreamgirls".
Presenters at the ceremony included Lin-Manuel Miranda, John Boyega, Nathan Lane, Denise Gough, Andrew Garfield and Maureen Lipman.