All he could think of in that moment was to thank as many people as he could in the short time designated for his acceptance speech.
"Don't curse, don't do anything that's going to embarrass your parents," Odom joked on "CBS This Morning." "I'm still taking in the moment. I came here straight from the after-party."
Odom's road to the Tony stage took a number of twists and turns along the way. At 13, he was mesmerized by a news report on the Broadway musical "Rent." He made it his life's ambition to be part of the show, and four years later, he scored a small part.
After finishing college, he pursued a career in television, but did not achieve the success he had hoped for.
When he was given the opportunity to read for Aaron Burr at a workshop, Odom made sure there was no question of how much he wanted the role. He memorized the entire part -- an uncommon act of dedication so early in the process. He said he knew he wanted the part "immediately" when he heard the musical's tracks.
"You hear a song like 'Wait For It,' you hear a song like 'Dear Theodosia' -- if you get one of those songs in a musical -- one -- it's worth dropping everything to sing that one song," Odom said. "It's the fact that... Aaron Burr gets 'Wait For It' and 'Dear Theodosia' and 'The Room Where It Happens,' and he gets to be the audience guide in that way. It's just such a tremendous responsibility."
"Hamilton" co-star and creator Lin-Manuel Miranda was also nominated for lead actor.
"I depend on Lin. He's half of my performance. Half of my performance I get from looking into his eyes," Odom said. "I wanted one of us to take it home."
On Sunday, the event celebrating and honoring the best in theater, television film and music also became a tribute to the victims of the horrific Orlando nightclub massacre. Odom had some initial reservations after reading posts on social media urging a postponement of the Tony Awards, but he changed his mind.
"I think that joy in the face of something like that is in its own way a protest," he said.
Odom is also living his dreams off Broadway. This month, he released his self-entitled debut album, a deeply personal project where he says he gets to play the true role of a lifetime -- himself. The album debuted at number one on the iTunes jazz albums chart.
"We want to pull out songs from the American song book and we want to make them palatable for a modern audience," Odom said. "We pitched the album saying we want to make the music that Nat King Cole would make today, so we want to modernize this sound and make this music relevant and important again."