It was almost 2 a.m. Monday when Orin Wolf, the producer of “The Band’s Visit,” stepped onto the dance floor at the Bryant Park Grill, a microphone in his left hand and a Tony Award in his right.

There was falafel and hummus on the buffet table, and the D.J. was playing a dance remix of “Sidi Mansour,” a Tunisian folk song. Ossama Farouk, a member of the show’s band, and his brother Sam were keeping time on the doumbek, a type of Egyptian drum, but they stopped when Mr. Wolf began to speak, trying to make sense of the musical’s Tonys sweep just hours earlier. (Standing 6 feet 5 inches tall, he’s easy to pick out in a crowd.)

“This show is about unity,” Mr. Wolf said, reflecting on the show’s depiction of Arabs and Israelis who unexpectedly find a common humanity over a single night, “and this award is about unity.” Then he asked the cast and crew to pose for a photo, and gave an instruction few producers ever have the opportunity to give: “Raise your Tonys in the air.”

The scale of victory by “The Band’s Visit,” a quiet musical in which not very much happens, has stunned Broadway. The musical won 10 awards — more than “A Chorus Line” or “The Lion King” or “The Book of Mormon” or “Dear Evan Hansen.”