The hit show about the life of one of America's founding fathers just fell short of The Producers' record of 12 statuettes in 2001.
For the first time at the ceremony, all four musical acting winners were black, including British star Cynthia Erivo.
But Saturday's mass shooting in an Orlando gay club overshadowed the show, and many paid tribute to the victims.
Host James Corden dedicated the night in New York to celebrating the diversity of Broadway, saying: "Our hearts go out to all of those affected by this atrocity. All we can say is you're not on your own right now. Your tragedy is our tragedy.
"Hate will never win. Tonight's show stands as a symbol and a celebration of that principle."
Corden won a Tony Award in 2012 for his performance in One Man, Two Guvnors.
US President Barack Obama and First Lady Michelle introduced the cast of Hamilton, which had 16 nominations, with a video message at the ceremony. The president said it was "not only a smash hit but a civics lesson our kids can't get enough of", while rapper Common described the show as a "cultural phenomenon".
It won awards including best musical, best director, best original score and best book of a musical for the spoken, non-sung dialogue and storyline.
The cast changed their performance in the light of the Orlando shooting, which killed 50 people and wounded 53, leaving their prop guns behind when they took to the stage.
Hamilton, which won the Pulitzer Prize for drama earlier this year, has been hailed as "a landmark American musical", garnering celebrity fans including Star Wars director JJ Abrams, Jay Z and President Obama.
In addition to blockbuster ticket sales, the album of the show is the highest-selling cast recording for 50 years, picking up this year's Grammy Award for best musical theatre album.
Erivo, who won best actress for her leading role in the Broadway revival of The Color Purple, said: "I promised myself I wouldn't cry because my make-up artist would be really mad at me. Thank you American Theatre Wing for making a London girl very, very happy."
The Color Purple began at the Menier Chocolate Factory in London in 2013, where Erivo played Celie. Its artistic director David Babani said: "It's an honour to be recognised in this way by the theatre community in New York.
"However wonderful the events of tonight, they are overshadowed by the tragedy in Orlando, and our thoughts and prayers are with everyone affected by this awful event."
The other three best musical actor winners were from Hamilton. They were Leslie Odom Jr for lead actor, Daveed Diggs for featured actor and Renee Elise Goldsberry for featured actress.
The show is expected to open at London's Victoria Palace Theatre in October 2017.
Speaking before the ceremony, Hamilton's star and creator Lin-Manuel Miranda said the shooting was "heavy in my heart and heavy in my mind".
Veteran US actor Frank Langella, who won best performance by a lead actor in a play for his role in The Father, urged Orlando to be "strong" in the wake of the atrocity.
"I'm standing in a room full of the most generous human beings on earth and we will be with you every step of the way," he said.
Jessica Lange won the award for best performance by a lead actress in a play for her role as morphine-addicted Mary Tyrone in Long Day's Journey Into Night.
"This is a dream come true. It fills me with such happiness even on such a sad day as this," she said.
Barbra Streisand also attended the ceremony for the first time in 46 years to hand out the best musical award.
Hip-hop musical Hamilton has led the Tony Awards in New York, winning 11 including best musical and lead actor.