Rashford scored two late goals as United saw off the visitors, who produced a resilient display to do their 9,000 travelling fans justice at Old Trafford.
The League One side had stayed in the game until then and had some moments themselves, with Scott Fraser missing a good chance to level at the start of the second half.
Their manager, Dean Holden, wants Charlton to use their display last night to kick-start what has so far been a difficult season.
Charlton are in 12th place in League One, nine points outside the play-off places, as they target a promotion push in the second half of the campaign.
“Pride in the performance and pride in the way we showed ourselves as a football club, on and off the pitch,” said the new boss.
“We rode our luck for the first 20 minutes, but we defended the box really well. Which club comes here and doesn’t have to ride their luck in the Premier League? We grew into the game on the back of that. I said to the players that I was really pleased with the belief we showed with the ball at our feet.
“We were so close. How many times on the counter-attack were we close to finding that last pass, that final moment? But we just couldn’t find it.
“We came out in the second half and had a big chance with Fraser, I thought we had a grip of the game at that point. I thought we were in control. I turned to my left and Erik is bringing those three on: Casemiro, Eriksen and Rashford.
“All in all, I thought the scoreline was harsh, to be honest with you. I’m disappointed with losing the game. We’re trying to get a winning culture into the team. Having said that, I am still very proud. This can’t be ‘after the Lord Mayor’s show’ now. We have got Barnsley at the weekend.”
Following the match, lifelong United fan Holden was invited into Ten Hag’s office. “Erik invited me in after the game and was talking about some of the quality that our players showed,” he said. “That gives me pride. I have so much respect for Erik. I have been watching from afar. I hope United win it. I want to take my kids to another cup final!”
Charlton owner Thomas Sandgaard was in the directors’ box at Old Trafford, as was the former Sunderland executive Charlie Methven, amid talk that he is involved in a takeover bid for the south-east London club.
Holden, who was appointed in December on a deal until the end of the season, thinks Charlton and their fans showed the potential of the club.
“I talked in the week about showing the best of ourselves on and off the pitch,” he said. “We knew what backing we were going to get and there’s a connection there, you can see that. Even with all the rumours off the pitch, the players have shown real focus. It hasn’t got to me as the manager of the club.
“Outside, what will be will be. It was important to use tonight to come out a better club and I think we have done that.”