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Producer Bernard Gersten added he wanted to thank the company for taking him "as close to theatrical utopia as we're likely to get".
Briton Jennifer Ehle was named best featured actress in a play for her role in Utopia.
As she collected her award, she said: "Oh my God, thank you so much.
"Thank you... for having the courage to do The Coast of Utopia and to show New York, New York's audiences and New York's actors, that we really do all want a repertory company at the Lincoln Centre, and hopefully one day that will happen."
Earlier, O'Brien paid tribute to Stoppard for giving him "the ride of my life" as he collected his best director award and said: "I know what Everest feels like."
Actress Marsha Gay Harden, who introduced clips from each of the best play nominees, described Utopia as "emotionally passionate and visually absorbing".
Utopia dominated the 61st annual Tony awards at the Radio City Music Hall on New York's Broadway, along with Spring Awakening, which picked up best musical.
Producer Ira Pittelman said: "Steven Sater, Duncan Sheik and Michael Mayer, we thank you for conceiving and creating a Broadway musical that rocks, that makes you laugh, and cry and think. A show that speaks from the heart and touches audiences of every age and every generation."
Spring Awakening also won the best score award for Duncan Sheik and lyricist Steven Sater, who also received the prize for best book of a musical.
Sheik, who also won best orchestrations, said: "Musical theatre rocks.
"Steven and I definitely set out to make a new kind of musical. We were trying to forge our own path. I think we got lucky timing-wise - what's happening politically. People were ready to deal with something that had teeth."
Its director, Michael Mayer, also won as did John Gallagher Junior, who portrays a manic student in the show.
Gallagher, 22, said: "Heaven must feel like this."
Bill T Jones danced down the aisle as he accepted his award for choreography for the musical.
"I am a happy man," he said.
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