"This means so much for a little Australian girl," she said at the ceremony at New York's Radio City Music Hall on Sunday (Monday AEST).
She thanked her husband, Dave Lawson, for "holding down the fort and keeping our family together".
Hours before the main even began Australian Marg Hornwell was announced as the winner of best costume design in a play for A Picture of Dorian Gray.
Snook plays all 26 roles in the one-woman The Picture of Dorian Gray, which was originally produced by Sydney Theatre Company and won the actress a prestigious Olivier Award in London.
Snook received international recognition for playing Shiv Roy in the TV series Succession, for which she won and Emmy.
The Picture of Dorian Gray has six Tony nominations, including best direction for Kip Williams with his fellow Australians Clemence Williams, Nick Schlieper, David Berman nominated for best sound, lighting, and set design - as well Hornwell's win for costumes.
Francis Jue won best actor in a featured role in a play for his work in a revival of Yellow Face.
He said he was gifted the tux he was wearing from another Asian actor who wanted him to wear it to the Tonys.
"I'm only here because of the encouragement and inspiration of generations of wonderful deserving Asian artists who came before me," he said.
"To those who don't feel seen," he added. "I see you."
Jak Malone won best actor in a featured role in a musical for the British import Operation Mincemeat: A New Musical, playing a woman every performance. He hoped his win could be a powerful punch for trans rights.
with AAP
