The Musical, a jukebox adaptation of Australian director Baz Luhrmann's hyperactive 2001 movie, earned 10 trophies, including best musical.
In a surprise to no-one, Aaron Tveit, who plays Christian, won the award for best leading actor in a musical — he was the only person nominated in the category.
Moulin Rouge also won for scenic design, choreography, costume, direction, lighting, sound design, orchestrations and a featured acting Tony for Broadway favourite Danny Burstein, who plays nightclub owner Harold Zidler.
The sobering musical about an American family spiralling out of control won for best book, and Lauren Patten won the award for best featured actress in a musical as Jo.
The Tony Awards traditionally comes with some A-game performances but Jennifer Holliday took it to a whole new level with her reprisal of And I Am Telling You I'm Not Going.
Adrienne Warren won the Tony for best leading actress in a musical for her electric turn as Tina Turner in Tina — The Tina Turner Musical.
The Inheritance also yielded wins for Andrew Burnap as best actor in a play, Stephen Daldry as best director, and Lois Smith as best performance by an actress in a featured role in a play.