The England captain converted a 13th-minute penalty to give England a 1-0 lead in Serravalle after the Cypriot referee, Leontios Trattou, blew for a foul on John Stones by Davide Simoncini.
The match ended 6-0 in favour of the visitors.
Rooney’s goal gave the 29-year-old Manchester United striker an identical international record to Charlton, a member of England’s fabled 1966 World Cup-winning team, with his 49th goal coming in his 106th appearance.
“To be level with Sir Bobby is a huge honour, a great moment and I am very proud of it,” he told BBC Radio 5 live.
“Hopefully I will surpass that and become the outright leader.”
He now requires one goal to become the first England player to score a half-century of goals.
Rooney is also closing in on Charlton’s United scoring record and requires 17 goals to surpass the former midfielder’s tally of 249.
But he is currently struggling for goals at club level.
Despite a hat-trick in a recent Champions League play-off win away to Belgian side Club Brugge, Rooney has not scored a league goal since April.