It received 11 nominations, including best musical.
The show, adapted from the hit 2006 indie film and featuring the Oscar-winning song ‘Falling Slowly’, also gained nominations for its main stars, Steve Kazee and Cristin Milioti.
Two other musicals — The Gershwins’ Porgy and Bess and Nice Work If You Can Get It — each got 10 nominations.
Once, with songs by Marketa Irglova and Glen Hansard, was originally a low-budget movie made for little over €100,000. The film earned $20m.
The musical is a study in how to adapt a movie to the stage.
Its director, John Tiffany, was at a play in Glasgow when his phone began buzzing “like crazy” with word of the show’s success. He sat through the play but learned about the Tony haul after glancing at texts.
“It’s just brilliant. It’s incredible because it’s such a delicate, unassuming show.
“But Once constantly surprises me. I think it’s the power of the music and the storytelling that people connect with,” said Tiffany, who was nominated for best director of a musical.
Kazee said: “I think it’s a universal message about love, not just between two people, but a love of music, a love of your country, how we share things as people.
“There are moments that are just snapshots in your life, but they stay with you forever. We all have those moments.”
The Tony Awards will be held at New York’s Beacon Theater on June 10.