0710 GMT May 19, 2019
The 29-year-old Manchester United striker's penalty in the 2-0 win over Switzerland in a Euro 2016 qualifier took his tally to 50 goals, BBC reported.
"It's a great feeling," he said. "I've been close for the last few games and it's a dream come true.
"I've got a lot of football in me for England. I hope the goals don't stop."
The spot-kick was awarded late in the game when Raheem Sterling was fouled in the area by Granit Xhaka.
Rooney, who scored his first England goal against Macedonia in 2003, told BBC Radio 5 live, "It's a very special night for me and my family. A night I will never forget.
"I was very nervous stepping up to take the penalty. I knew it was such a big moment for me and my career. The players said to me afterwards it was like being at a concert because there were so many flashes from phones in the crowd.
"It's a huge honor because Sir Bobby Charlton is an England and Manchester United legend. I know Sir Bobby will be extremely proud that a Manchester United captain has broken the record."
The striker was handed a commemorative England shirt by boss Roy Hodgson in the dressing room after the match, and made a speech in front of his colleagues.
"Nights like this you don't forget," he added. "When you create a bit of history it's a great feeling. It was really nice of the players and the manager to present me with a shirt afterwards."
England coach Roy Hodgson praised Rooney's achievement, and said the former Everton forward could possibly reach a tally of 65 goals.
"You would like to think we are on about 10 games a year and maybe if we ever did very well in a tournament a bit more — that is at least to 2018," he said.
"So you are talking 20, maybe even 30 games, so it could be like 10 to 15 goals.
"I know he is really delighted. He will tell you what a proud moment it is for him and his family. I also know, as a team man, he will be thinking 'how can I help England become a better team?', 'Can I make certain — like Bobby Charlton — I end up as a winner in a tournament or help England do very well?'
"You can't do much better than be number one in the goal scoring list, but I still want to say a word for Bobby Charlton. He is obviously a legend and one of the finest players I ever saw and I don't think we will diminish or debase Bobby's legendary status.
"With him and Wayne Rooney we are talking about two of the finest football players in England and I'm lucky enough to have one still playing and that I can use him."