The Rocket Man singer, 63, and partner David Furnish, 48, became parents when the little boy was born via a surrogate on Christmas Day.
Speaking to OK! magazine about becoming a dad, Elton said: "There just aren't proper words to describe that feeling. It's indescribable. It was just... We have a son! We have a son!"
The couple said they were 'heartbroken' when they were unable to adopt Ukrainian orphan Lev, 14, months, and his three-year-old brother Artyom in 2009 because of legal matters.
Despite not giving up on the brothers, the couple then turned to surrogacy in their bid to become parents.
Elton said: "On the Christmas before last, I remember we were in the kitchen at Woodside.
"We'd realised that we would not be able to adopt Lev and Artyom, and then David said: 'Well, what about surrogacy?' I said: 'You know what? Why not?' I'm 62 at this point, but I feel 40. I said: 'We could give a child a wonderful home.'"
The pair then used an agency called the Centre For Surrogate Parenting where their surrogate picked them out anonymously.
David said: "We looked through a whole series of egg donor profiles, with the primary concern being family health history. Then our surrogate chose us from an anonymous profile - it's her choice, too."
He added: "Initially, we used pseudonyms and got to know her by telephone. Then one day, she just figured out who we were. She cracked it. And it didn't make a bit of difference to her."
To keep the whole pregnancy secret and to keep the surrogate anonymous Elton was not able to visit her.
He said: "I couldn't be seen anywhere near her for confidentiality reasons. But we spoke on the phone, and David and I communicated like we normally do, eight or nine times a day. And eventually I met her, and we got on like a house on fire."
Elton said their little boy was finally born on December 25 after the surrogate went through a 12-hour labour at LA's Cedars Sinai hospital.