Rooney, 31, broke the deadlock in the seventh minute at Old Trafford, steering Juan Mata's cross over Reading goalkeeper Ali alHabsi with his right knee.
Former United midfielder Charlton, 79, and former United manager Alex Ferguson were both in attendance and joined fans in applauding Rooney's feat.
Guardiola gets first taste of FA Cup
Ex-United captain Bryan Robson was one of the first figures from the club to offer his congratulations, writing on Twitter: "Congratulations @WayneRooney for equalling Sir Bobby's record."
Rooney, who was making his 543rd appearance for the club, broke Charlton's England scoring record of 49 goals in October 2015 and has now scored 53 times for his country.
Charlton's United appearance record was broken by Ryan Giggs in May 2008.
Rooney joined United from Everton at the age of 18 in 2004, after starring for England at Euro 2004, and scored a Champions League hat-trick against Fenerbahce on his debut.
United hope FA Cup defence won't come at a cost
He has since won five Premier League titles, two League Cups, last season's FA Cup, the 2007-08 Champions League and the 2008 Club World Cup.
He was voted the Professional Footballers' Association Player of the Year and the Football Writers' Association Footballer of the Year in 2010.
Rooney was named United captain by former manager Louis van Gaal in July 2014 and was given the England captaincy by then manager Roy Hodgson a month later.
Wenger demands more FA Cup success
But he has lost his place in United's starting XI following Jose Mourinho's arrival as manager and has had to make do with occasional starts in the cup competitions of late.
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