The awards, hosted this year by James Corden, celebrate the best of Broadway in New York.
Stroker, who was paralysed in a car crash when she was two, dedicated her prize to children with disabilities or facing limitations as she was named best featured actress in a musical for her performance as Ado Annie in the revival of Oklahoma.
Hadestown was the big winner at the ceremony, taking home eight prizes including best musical.
The award for best play went to Northern Irish drama The Ferryman, while British director Sam Mendes also picked up his first Tony for directing.
Accepting her award as the crowd erupted into applause, Stroker said: “This award is for every kid who is watching tonight who has a disability, who has a limitation or a challenge, who has been waiting to see themselves represented in this arena – you are.”
The Ferryman, written by James Bond co-writer Jez Butterworth, tells the tale of a former IRA man in rural County Armagh during The Troubles.
It was transferred to Broadway with much of the same cast after a sold-out year in the West End.
Accepting the award, Butterworth dedicated the prize to the families who had lost loved ones in the Troubles.
Mendes did not attend the ceremony, but posted a message on The Ferryman’s Twitter page, saying: “I have lost count of the times on this job that I’ve counted my lucky stars.
“And now, again, I’m left to reflect on what a privilege – what an absolute privilege it is – to be able to tell stories for a living.”
Laura Donnelly – whose family story inspired the play – was beaten by Elaine May in the best actress in a leading role category, while Paddy Considine and Fionnula Flanagan also missed out on gongs.
British comedian Corden kicked off his hosting duties with a nine-minute musical tribute to Broadway.
Other winners included English actor Bertie Carvel, who picked up best featured actor in a play for his portrayal of Rupert Murdoch in Ink, and Breaking Bad star Bryan Cranston, who scooped the lead actor in a play award for his performance in Network.