Rooney, 31, scored his 249th goal for the club to match Bobby Charlton's record, set in March 1973, as United made it eight wins in a row — their best run since September 2009.
Rooney could now break the record against Hull City in Tuesday's League Cup semi-final first leg and United manager Mourinho said: "I think the best day for him will arrive. His best day will be the next goal. If he scores the next goal, he becomes the top scorer in the history of Manchester United."
Rooney equals Charlton's Man Utd scoring record
The Portuguese compared his captain with the legendary Charlton, saying: "It's amazing because everyone knows who Sir Bobby is and what he means for the club. So for Wayne to score as many goals as him for Manchester United is fantastic."
Rooney turned provider to set up Anthony Martial for United's second goal at Old Trafford in Saturday's third-round tie before Marcus Rashford broke his 17-match goal drought with a second-half double.
Rooney was delighted to make history after matching a record that has stood for almost 44 years, particularly with Charlton watching on from the directors' box.
"This club is a huge part of my life and I'm honoured to be up there alongside Sir Bobby," he said. "It's a proud moment. To do it at a massive club like Manchester United, I'm hugely honoured."
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Reading boss Jaap Stam, making his first return to Old Trafford as a manager after his hugely successful time as a United player from 1998 to 2001, also paid tribute to Rooney.
"I think Wayne has been a great player, right from the beginning when he arrived here," said the 44-year-old Dutchman. "He's an outstanding player and it's no surprise he has scored so many goals. You can see what it means to him because even when they were 4-0 up, he was still running."
United were ruthless and could have scored many more as they strolled into the fourth round with a routine win.
"It was very professional — I could see that from the first minute because we had two chances to score," said Mourinho. "We played well, but we knew they were good with the ball so when we gave them the chance with the ball, they moved it well and created some things.
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Mourinho also hailed Rashford, who ran through to score from Michael Carrick's pass in the 75th minute before pouncing on Reading goalkeeper Ali Al Habsi's woeful miskick to net his second goal four minutes later.
"I spoke with him at half-time and told him I don't care with the goals he missed in the first half because he had a big responsibility in the way we were breaking their defensive line," said Mourinho.
"He was playing so well. One more goal, one less goal, it doesn't change. But for an attacking player to score a goal means happiness, confidence. It was good for him.
Stam admitted it was a learning curve for his second-tier side, who never seriously threatened United's goal despite a spirited spell in the second half at 2-0.
"It's not always a good experience when you win games, but it can also be when you lose games as well," he said. "You see what you need to improve to make yourself better."
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