The landmark came in the seventh minute at Old Trafford when Rooney steered Juan Mata's cross past goalkeeper Ali Al Habsi with his right knee.
Charlton, 79, and the former United manager, Sir Alex Ferguson, were among the supporters who applauded after the England skipper achieved the feat.
"It's a proud moment and to do it at a massive club like Manchester United," Rooney told BT Sport. "I'm hugely honoured to play for this club. To be up there with Sir Bobby in terms of goals is a huge honour. Hopefully I'll be up there on my own soon.
"You want to do records in front of our home fans. We've got two home games coming up and hopefully I can get the record in one of them."
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 in 119 games.
Charlton's United appearance record was broken by Ryan Giggs in May 2008. "Bobby Charlton is Bobby Charlton. But to have Wayne as the top scorer in such a big club is great for him," said United manager Jose Mourinho. "He's an amazing guy in the group. We all want him to do it. It's amazing, but I want one more goal, a very important one, a last minute goal."
Rooney joined United from Everton at the age of 18 in 2004. 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 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.
But under Mourinho he has not been an automatic selection in the United line-up for Premier League games.