Giles Terera won best actor in a musical for his portrayal of Aaron Burr while his co-star Michael Jibson won the best actor in a supporting role in a musical award.
Terera said: "I am relieved and elated that we did so well.
"We did not take anything for granted. You never know how these things will be received and I knew it was very special."
Asked if the production was a "game changer", Terera said: "Yes, I think so."
The hip hop show tells the story of American founding father Alexander Hamilton.
The musical's creator Lin-Manuel Miranda was awarded the outstanding achievement in music trophy along with the show's orchestrator Alex Lacamoire.
Bryan Cranston who won best actor for his portrayal of Howard Beale in Network also wore a pin on the red carpet.
He said: "The idea that older white men are controlling the world and having free rein is over.
"With every person that is brought to the attention, and the aggressors, whether sexually or power oppressors, when they fall, we have the opportunity to rebuild on a foundation of mutual respect. Invisible of gender, of sexual preference, of colour, let's build it up with mutual respect of everyone.
Who were the main winners of the night?
Best new musical
Best actor in a musical
Giles Terera (Hamilton)
Best Actress in a musical
Shirley Henderson (Girl From The North Country)
Best actor in a supporting role in a musical
Michael Jibson (Hamilton)
Best actress in a supporting role in a musical
Sheila Atim (Girl From The North Country)
Best new play
Bryan Cranston (Network)
Laura Donnelly (The Ferryman)
Sam Mendes (The Ferryman)
Best actor in a supporting role
Bertie Carvel (Ink)
Best actress in a supporting role
Denise Gough (Angels In America)
Best new comedy
Labour Of Love
Best new dance production
Outstanding achievement in dance
Francesca Velicu (Le Sacre du Printemps)
Best entertainment and family
Best costume design
Vicki Mortimer (Follies)
Best sound design
Nevin Steinberg (Hamilton)
Outstanding achievement in affiliate theatre
Best set design
Bob Crowley (An American In Paris)
Best lighting design
Howell Binkley (Hamilton)
Best new opera
Outstanding achievement in opera
Joyce DiDonato and Daniela Barcellona (Semiramide)
Angels In America
Outstanding achievement in music
Best theatre choreographer
Andy Blankenbuehler (Hamilton)
Best musical revival