The former Breaking Bad star won the prize for Best Lead Actor in a Play at the ceremony, which took place at New York's Radio City Music Hall and aired live on CBS.
"Oh, my goodness! Oh, dear lord! My first Broadway play was in 1977. I snuck into the second act of Hair and, to this day, I still haven't seen the first act. But they tell me the second act is better because there is more nudity, so I feel blessed. ...
"To all the fellow actors in this category, I am deeply honored to be in your company," he told his Tony rivals Samuel Barnett, Mark Rylance, Chris O'Dowd and Tony Shalhoub.
"Thank you so much for this," he addressed the crowd, cradling his trophy and referring to the script that helped him earn it as a "beautiful and important play."
Moments later, Robert Schenkkan's drama scored the Tony for Best Play.