Celebrities have been flocking to see the Tony-nominated play, which documents the life of U.S. Founding Father Alexander Hamilton through rap music.
U.S. President Barack Obama made sure to take in the show during a recent visit to New York, but it was Spielberg's presence one night which really made Miranda step up his game.
He told U.S. breakfast show Good Morning America, "It's thrilling (to have high-profile guests in the audience) and sometimes, I look out and, 'Hey, Steven Spielberg! I see you on the aisle!'
"All I wanted to do was be Steven Spielberg when I was a kid. I just thought, 'OK, Steven Spielberg's watching me. I guess I'm in a Steven Spielberg movie in my own mind for the next two hours and 45 minutes!' That was one of the best shows I ever did, because I was trying to be worthy of the audience that I had, and I kind of feel that way every night."
The musical, which is sold out night after night, will compete for a record-breaking 16 Tony Awards at the annual ceremony next month (Jun16). It's up for Best Musical, Best Choreography, and Best Performance, while Miranda is the favourite to land the Actor in a Leading Role in a Musical prize - and the critical success has left the star stunned.
"I didn't know that was a thing that was possible," he gushed about Hamilton's 16 nods. "It's been really overwhelming. It takes a lot of people to put on a show. For everyone in the departments to be recognised is a validation of our work."
Meanwhile, Miranda is hoping to one day adapt his show into a film, although it won't be anytime soon.
Asked if a big screen version is something he is interested in, he replied, "Later. Honestly, I don't know how much later, but I worked seven years to make this thing a piece of theatre, so I would obviously love for as many people as possible to experience it as a piece of theatre first and I'm in no rush (to turn it into a movie)."