"When senseless acts of tragedy reminds us that nothing here is promised. Not one day," he said in reference to the Orlando massacre. "This show is proof that history remembers, we live through times when hate and fear seem stronger, we rise and fall and light from dying embers, remembrances that hope and love last longer."
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He added: "Love is love is love is love is love, and love cannot be killed or swept aside. Fill the world with music, love and pride."
Earlier in the evening, host James Corden also honored the victims of the gay nightclub shooting in Orlando which killed at least 49 people. The 29-year-old gunman, an American citizen, was shot and killed by police.
"Hate will never win," the "Late Late Show" host said. "Together, we have to make sure of that. Tonight's show stands as a symbol and celebration of that principle."
Actor Frank Langhella, while accepting his fourth Tony for best actor in "The Father," also brought up "a dose of true reality-what happened today in Orlando."
"I urge you Orlando to be strong," he added. "I'm standing in a room full of the most generous human beings on earth and we will be with you ever step of the way."
Miranda's acclaimed musical, which tells the story of Founding Father Alexander Hamilton through hip hop and a multicultural cast, took out 11 awards, including Best Musical.
It won for best leading actor (the Tony went to Leslie Odom Jr. who played Aaron Burr, the rival to Hamilton, played by Miranda), best original score, best book of a musical, best choreography, best direction, best orchestration, featured actor and actress for Renee Elise Goldsberry and Daveed Diggs, best costume and best lighting. Miranda was nominated for leading actor but lost to Odom Jr.
The musical fell short of one award to break the record; it was nominated for a record 12 nominations and garnered 11.
The Tonys showcased a year of diversity: 14 out of the 40 Tony nominees for acting in plays and musicals, a total of 35 percent, were actors of color.
At one point in the show, the crowd, including Miranda, danced with Gloria Estefan and Ana Villafañe, who plays Estefan in the hit musical "On Your Feet." Onstage before the "On Your Feet" medley began, Emilio Estefan joked that all the Latino cast of the musical was in the country legally, in a jab at Trump.