Antony’s superb first-half opener gave United the advantage but Dean Holden’s visitors claimed plenty of credit for the way they continued to battle until second-half substitute Rashford found the net for a sixth successive game with goals in the 90th and 94th minutes.
Charlton’s 9,000-strong travelling support, their largest ever for a midweek cup tie, applauded their side off, having backed them throughout, briefly raising hopes of an equaliser when Scott Fraser curled his free-kick straight at Tom Heaton.
For United, it means eight wins on the trot and a third last-four appearance in this competition in six seasons since they won it in 2017 under Jose Mourinho, their last domestic silverware.
They lost the last two to Manchester City, who United face in the Premier League at Old Trafford on Saturday (12:30 GMT).
United boss Erik ten Hag said after the game that he was “quite happy” with the first half an hour but was “was less happy we were not clinical in front of goal”.
“We had to be in that semi final and we did the job,” said the Dutchman.
“I think we had to finish this off early on but in the end the objective was to get through so compliments to the team.”
Eight on the trot for in-form United
Eight successive wins for the first time since Ole Gunnar Solskjaer made such a brilliant start to his tenure in the middle of the 2018-19 season is a decent way for Ten Hag’s side to approach the latest Manchester derby.
This triumph was achieved with a team missing at least seven players expected to be part of United’s starting line-up against City, with expectations high they will avoid the chastening first-half endured at Etihad Stadium, when they trailed 4-0.
The progress since then has been immense. And, in Antony, they have an £81.3m winger who has now scored in successive games and brings trickery to wide areas straight out of the United tradition.
The Brazilian’s first-half opener owed much to Diogo Dalot. The Portugal full-back did not last the opening period due to injury but it was his off-the-ball burst into the Charlton penalty area that created the space for Antony to curl a superb effort from Fred’s square pass into the far corner.
Fred smacked a thunderous shot against a post and Christian Eriksen brought an excellent save out of Ashley Maynard-Brewer before Rashford extended his scoring streak.
First he swept home a pass from debutant Facundo Pellistri before he beat Maynard-Brewer in the final minute of stoppage time with a precise finish.
The England man is arguably in the best form of his career and Pep Guardiola already knows the damage he is capable of inflicting on a City defence that will need to be on its guard every bit as much as their hosts’.
Proud Charlton need some answers
This performance was a reminder of what Charlton could be.
The 9,000 travelling fans provided plenty of backing for Holden’s men and they were still in the game as the clock ticked toward injury time.
Alongside owner Thomas Sandgaard among the Charlton delegation in the directors box at Old Trafford were former Sunderland executive Charlie Methven and Simon Lenagan, son of Wigan Warriors owner Ian.
Both were listed as ‘club guest’ and both have been to recent Charlton matches, fuelling suggestions they are part of the secretive group that has apparently been granted a period of exclusivity around the ownership, even though Dane Sandgaard says he is going nowhere.
It is the kind of situation Charlton fans thought they had seen the back of when they fought off one potential owner they did not want in 2020, and why they feel an Independent Regulator for the game as a whole is needed as a matter of urgency.
Not that they were thinking too much about that in the opening minute of the second-half when Corey Blackett-Taylor lined up Charlton’s best opportunity to equalise, only to fire over from the edge of the area.
Fraser had their only other chance and despite the late setbacks, there was passion and admiration in the ovation the supporters offered Holden and his players at the end.
Now they return to more mundane fare, wanting to know who is running their club.
After the game, Holden said: [He felt] pride in the performance, the way we showed ourselves as a club on and off the pitch.
“We rode our luck in the first 20 minutes and grew into the game after that. We were so close on how many times on the counter attack to find that last pass and just couldn’t find it.
“We came out in the second half and had the big chance with Fraser and I turned to my left and Erik brings them three on, Casemiro, Eriksen and Rashford. I thought the scoreline was harsh.”
He added: “You can only stand in admiration for Casemiro’s pass for the third goal, it was like watching Steven Gerrard and Rashford does what Rashford does at the end.”