The two-part spectacle captured six accolades, including best play, book, lighting, sound design, orchestrations and director for John Tiffany, who asked the crowd to sing Happy Birthday to his boyfriend.
The night also saw British actor Andrew Garfield win his first Tony, for best leading actor in a play, for playing a young gay man living with Aids in the sprawling, seven-hour revival Angels In America.
Garfield dedicated the win to the LGBTQ community, who he said fought and died for the right to love. He said the play is a rejection of bigotry, shame and oppression.
“We are all sacred and we all belong,” Garfield said.
Garfield then referenced last week’s US Supreme Court decision, which ruled in favour of a baker’s right to deny a gay couple a wedding cake based on his beliefs.
“(Let’s) just bake a cake for everyone who wants a cake to be baked,” he said, to rousing applause.
Based on an original new story by JK Rowling, Jack Thorne and John Tiffany, Harry Potter and the Cursed Child is a new play by Jack Thorne.
It is the eighth story in the Harry Potter series and the first official Harry Potter story to be presented on stage.
Harry Potter and the Cursed Child is one play presented in two parts, which are intended to be seen in order on the same day (matinee and evening) or on two consecutive evenings.
Sunday night also saw Robert De Niro take to the stage to introduce a performance from Bruce Springsteen, with an expletive directed at US president Donald Trump.
“I’m gonna say one thing — ‘f*** Trump’,” he said, which garnered him a sustained standing ovation from the crowd.
In other wins of the night, Glenda Jackson scooped a Tony Award for best actress in a play for her work in a revival of Edward Albee’s Three Tall Women.
That show also yielded the featured actress win for Rosanne star Laurie Metcalf.
Composer Andrew Lloyd Webber and American actress Chita Rivera received honorary tributes, the Special Tony Award for Lifetime Achievement in the Theatre.
Thanking the audience, Lloyd Webber said: “I have to say, I am completely overwhelmed.”
He added: “What made me absolutely certain I wanted to be a musical theatre composer was the movie South Pacific. I devoured absolutely everything I could find by Rodgers and Hammerstein and all I wanted when I was ten years old was to be Richard Rodgers.
“I never dreamed that one day I, a Brit of all things, would be honoured with the same lifetime award that my idol won, by my peers and the true home of the musical, Broadway. I’m absolutely humbled.”
US musical The Band’s Visit was the other big winner.
The Band’s Visit, based on a 2007 Israeli film of the same name about an Egyptian band that goes to the wrong Israeli town, won 10 statuettes - including best musical, best direction, orchestration, sound design, best book of a musical, lighting and featured actor Ari’el Stachel, who gave a heartfelt speech about his past.
“For so many years of my life I pretended I was not a Middle Eastern person,” he said, addressing his parents in the audience.
He thanked the creators of the show “for being courageous for telling a small story about Arabs and Israelis getting along at a time that we need that more than ever”.
Angels In America star Nathan Lane won his third Tony, for best featured actor in a play
Lane said the play still speaks to society in the midst of “political insanity”.
In a mesmerising moment, Melody Herzfeld, the heroic drama teacher who nurtured many of the young people demanding change following the February school shooting in Parkland, Florida, was honoured from the stage.
Ms Herzfeld saved 65 lives by barricading students into a small classroom cupboard on Valentine’s Day when police say a former student went on a school rampage, killing 17 people.
The night also saw Billy Joel gave his friend Springsteen a special Tony Award.
“This is deeply appreciated, and thanks for making me feel so welcome on your block,” The Boss said.
Co-hosts Josh Groban and Sara Bareilles kicked the show off with a self-parodying duet on piano for all the losers out there - including themselves.
Neither Bareilles nor Groban have won a Grammy or a Tony despite selling millions of albums and appearing on Broadway. They turned that into a playful song.
“Let’s not forget that 90% of us leave empty-handed tonight. So this is for the people who lose/Most of us have been in your shoes,” they sang in the upbeat opening number. “This one’s for the loser inside of you.”