|Magical win: Hermione Grainger actress Noma Dumezweni and Anthony Boyle, who plays Scorpius Malfoy|
The wizarding saga was named best new play at the ceremony at London's Royal Albert Hall.
Best actor winner Jamie Parker, who plays the grown-up Harry, thanked the show's "resolutely enchantable audience".
Noma Dumezweni, who plays Hermione Grainger, won best supporting actress.
Oliviers 2017: The winners in full
Picking up her statuette, she thanked JK Rowling for creating the character. "The privilege of being able to play Hermione is overwhelming and humbling," she said.
The best supporting actor prize went to Anthony Boyle, who was plucked from acting school to play the role of Scorpius Malfoy.
Other highlights from Sunday night's ceremony include:
- Billie Piper winning best actress for Yerma
- Groundhog Day named best new musical
- Jesus Christ Superstar collecting the award for best musical revival
- Sir Kenneth Branagh honoured for his outstanding contribution to British theatre
Winning nine of its 11 nominations, Harry Potter and the Cursed Child overtakes Matilda the Musical and The Curious Incident of the Dog in the Night-Time - both of which had seven wins each - to become the most decorated production in Oliviers history.
The play opened to five-star reviews at the Palace Theatre last July and has already dominated theatre awards season.
Based on an original story by JK Rowling, Thorne and director John Tiffany, the two-parter presents Harry, Ron and Hermione in their mid-30s as their own children head off to Hogwarts school.
Tiffany was named best director - he was up against himself in the same category for The Glass Menagerie - while the play also won for its lighting, sound, costumes, and set design.
"The power of imagination is infinite," Tiffany said in his acceptance speech. "We need it now more than ever."
|Billie Piper will reprise her Yerma role at the Young Vic this summer|
Billie Piper's best actress win was the first award of the night.
In Yerma, which was also named best revival, she played a young woman driven to extremes by her desire to have a child.
"This is so cool, it's awesome," said the former Doctor Who actress, who will reprise her role at London's Young Vic in the summer.
National Theatre of Scotland's production of Our Ladies Of Perpetual Succour, which opens in the West End next month, was named best new comedy.
Matthew Bourne was named best theatre choreographer, with his production of The Red Shoes also winning best entertainment and family.
As well as its new musical prize, Groundhog Day - written by Tim Minchin and Danny Rubin - also received the award for best actor in a musical for Andy Karl's portrayal of Phil Connors. The Old Vic production opens on Broadway later this month.
|Dream duo: Amber Riley and co-star Adam J Bernard|
Former Glee star Amber Riley was named best actress in a musical for her role in Dreamgirls, while her co-star Adam J Bernard won best actor in a supporting role in a musical.
Picking up her award, Riley said: "Anyone who sees a little bit of themselves in me and sees me standing on this stage, I hope that you are inspired, I hope that you know that you can do this if you really want to."
Rebecca Trehearn won best actress in a supporting role in a musical for Show Boat.
|David Fynn (Dewey Finn) and the young cast of School of Rock|
Two shows composed by Andrew Lloyd Webber also scooped awards.
School of Rock the Musical won outstanding achievement in music for the three children's bands who play their instruments live every night at New London Theatre.
Meanwhile, Jesus Christ Superstar - first performed in 1971 - was named best musical revival.
The night included performances from Gary Barlow and Tim Firth with the company of The Girls and the Women's Institute Choir, as well as Amber Riley, Tim Minchin and the companies of School of Rock the Musical, Jesus Christ Superstar and The Red Shoes.
|Sir Kenneth Branagh was presented with a Special Award|
At the end of the night, Sir Kenneth Branagh was presented with a Special Award by actor Mark Rylance and was joined on stage by 60 of the actors who appeared in the recent Kenneth Branagh Theatre Company season of plays in the West End.
"There's nothing special about me," quipped Sir Kenneth. "I'm a working class boy from Belfast, and I was a comprehensive school kid in Reading."
The ceremony, hosted by Jason Manford, will be broadcast on ITV on Tuesday 11 April at 20:00 BST.