"We thought it would be great and we want to keep the film in people's consciousness, because it came out early and it's awards season," John told ET's Kevin Frazier at the event. "We thought it would be a great, fun way for people to reconnect with the film and the music because the music is a big part of the film."
As the credits rolled after the film, the duo performed "(I'm Gonna) Love Me Again" and concluded the evening with John on piano and Egerton singing "Don't Let the Sun Go Down On Me." John's longtime songwriting partner Bernie Taupin, as well as Rocketman stars Jamie Bell and Bryce Dallas Howard, and director Dexter Fletcher were also on hand.
"It's been a very surreal process getting to know him," Egerton said of the musical icon, adding that he admires "his capacity for love and warmth and his kindness, and his un-ended interest in me as an artist, not just in this project and how normal he's become to me."
"He doesn't want me to be constantly looking at him like he's God," the actor continued. "He wants me to look at him like he's my friend, which he is."
As for John, he's "so fond" of Egerton and considers him "part of my family." "I don't think I have to say anything to him because he seems to have it figured out for himself," John said of giving the actor career advice. "He's much more happy within himself and I think that's the truth."
It's been a busy year for John, who also released his his memoir, Me: Elton John, this week. In it, he opens up about his friendship with the late Princess Diana, and her two sons, Prince William and Prince Harry.
"Every time I do see them it brings back the fondest memories of their mother. They're both fantastic boys," the singer said. "They remind me of Princess Diana, and Harry...has his mother's ability to walk into a room and disarm people straight away and make them feel at ease, and that is a great gift."
The book also touches on John's memories of Michael Jackson, which included some of the hardest moments of the late singer's life, before his death.
"I knew Michael since he was 13, 14 years old," John explained. "What happened to him was such a tragedy, with the drugs and I don't think he had a particularly happy life leading up to his big success. I think success is hard to deal with and I think Michael found it hard and became isolated."
"It was awkward to be around him. It was tragic to be around him," he continues. "This was someone who was one of the most talented people to come on Earth, and it was so sad to see him destroyed by the drugs...He was a walking drug addict and was on everything possible. He was still sweet and lovely, but it's upsetting when you see someone that you care about and you can't do anything about it."