It equals Matilda's 2012 record for most Olivier Awards won by a musical.
The West End production of Lin-Manuel Miranda's show about US founding father Alexander Hamilton won best new musical and outstanding achievement in music.
But its creator had to stay away from the London awards, the biggest event in the UK theatrical calendar, having been diagnosed with shingles.
It was also a good night for The Ferryman, which picked up three awards.
The drama, set in rural Londonderry in 1981, against the backdrop of The Troubles, won best new play and best director for Sam Mendes, while Laura Donnelly was named best actress.
Miranda was jointly awarded the outstanding achievement in music prize with orchestrator Alex Lacamoire.
He sent Lacamoire a list of thank yous - set to the music of My Favourite Things - to sing to the audience at the show, at the Royal Albert Hall, on his behalf.
Giles Terera, who won best actor in a musical for his role as Hamilton's nemesis Aaron Burr, said he was "relieved, elated and happy" that Hamilton had done so well.
"We didn't take anything for granted," he said. "But I'm glad we got recognised."
Jamael Westman, who stars as Alexander Hamilton, said the show - which opened in London late last year - was "breaking down barriers, it's breaking down the understanding we have of musical theatre".
He added: "This is re-imagining what it can be, using the hip-hop form and more contemporary types of song and the use of language that is almost Shakespearean.
"It's changing the way that we see theatre and is breaking boundaries. That's why it's been recognised."
Hamilton's other awards include best supporting actor in a musical Michael Jibson, who plays King George III, as well as best theatre choreographer, best lighting design and best sound design.
Despite being the biggest winner of the night, it failed to break the record for most Olivier Awards set by Harry Potter and the Cursed Child last year when it picked up nine.
Cameron Mackintosh and co-star Michael Jibson, winner of the best actor in a supporting role
Picking up the best new play award, Butterworth said: "It's her story. I stayed in with a gin and tonic and she had to go out to work. I knew how much that cost her and how much this will mean."
After picking up her own award, Donnelly - who played widow Caitlin in the original production - said: "I'm thrilled - but I'm in shock. It's what I dreamt of from childhood so to get to this point is surreal."
She said because of the subject matter, she felt a "weight of responsibility I hadn't in any other job to do it justice, because it was so close to home."
The couple are taking The Ferryman to Broadway later this year.
Breaking Bad's Bryan Cranston took the best actor prize for Network for his portrayal of Howard Beale in the National Theatre production.
In his acceptance speech, he said he was grateful to be working in a country that funds the arts, adding: "Hopefully that will continue."
Speaking backstage, he added: "To support children's imagination and their ability to grow in a social, emotional way, [the arts are] more important than learning the dates of a war."
Other highlights of the ceremony, hosted by Catherine Tate, included:
Denise Gough was named best supporting actress for Angels in America.
Gough was not there to collect her award, but in a statement read by actress Ophelia Lovibond, she highlighted the importance of the Time's Up movement. Several actresses brought feminist activists as their guests, and many others wore Time's Up and 50/50 pins or rings to show their support.
Bob Dylan-inspired musical Girl from the North Country was also honoured, with Shirley Henderson winning best actress in a musical and Sheila Atim winning best supporting role in a musical.
Atim seemed genuinely shocked by her win.
"Guys, guys - this is mental!" she said.
One of the first people she thanked was Bob Dylan, "for writing this stuff before I was born".
'More women who look like me'
She then thanked women of colour who had won Olivier Awards before her.
"To me, it is very, very important. I really hope - especially with Time's Up and the 50/50 movement - that there will be more women who look like me up here."
Shirley Henderson said it felt "crazy" to win, and that the script for Girl From the North Country had been "everything I'd been looking for, for a long time" - having not been on the stage for 17 years.
David Lan won the special award in recognition of his outstanding contribution for his artistic directorship of the Young Vic.
Performances included one from past stars of Joseph and the Amazing Technicolour Dreamcoat - including Jason Donovan and Lee Mead - to mark the 50th anniversary of the Andrew Lloyd Webber and Tim Rice musical.