Rashford notched two goals in stoppage time to get on the scoresheet for the sixth successive match as the England forward killed the game after Antony's first-half strike had put United in front.
It was their eighth successive victory going back to early November as they booked a third League Cup semi-final in four years.
Demanding boss Erik ten Hag will not be handing out too much praise after this showing, though, as his much-changed side were pushed hard by the third-tier Addicks at Old Trafford.
Indeed, it was only after Ten Hag had introduced big guns Rashford, Casemiro and Christian Eriksen that they were able to seal their victory at the death.
Although they have won their last eight, and 11 of their last 12, a much sterner test comes on Saturday in the shape of the Manchester derby and they will need to be much better.
Still, they will be happy to be in the last four of the competition that represents their last domestic silverware in 2017 and one of Newcastle, Nottingham Forest, Wolves, Southampton or Manchester City will be up next.
The Addicks, managed by diehard United fan Dean Holden, can take pride in their performance and as they now concentrate on mounting a League One play-off push in the second half of the season.
It might have been even more convincing for the hosts had they taken one of the early chances that the prodigious talent Alejandro Garnacho created.
The Spaniard made an opening for Diogo Dalot after cutting inside from the byline, but the defender blazed over from a good position.
Garnacho then twice tried to go on his own as he had his marker Sean Clare on toast, first cutting in and then fizzing a shot just wide before a jinking run into the penalty area ended with him dragging a shot wide.
But it was from the right where United forged their 21st-minute opener.
They worked the ball well out to Antony, who made space on his preferred left foot and then sent a curling effort past Ashley Maynard-Brewer and into the top corner.
The hosts were within inches of doubling their lead 10 minutes before the break when Fred's 30-yard free-kick crashed into a post, with Maynard-Brewer a bystander.
Charlton, roared on by 10,000 travelling supporters, had battled hard and enjoyed their best period of the game heading into the interval as Albie Morgan whipped a free-kick wide and then Tom Heaton had to claim Clare's low cross with Corey Blackett-Taylor waiting to pounce.
They started the second half strongly as well and their golden opportunity came 76 seconds after the restart as Scott Fraser latched on to Miles Leaburn's flick-on but put his left-footed shot over.
Anthony Elanga had a goal ruled out for a clear offside but it was Charlton who continued to press their illustrious hosts and, such was their impact, Ten Hag introduced Rashford, Casemiro and Eriksen on the hour mark.
That experienced trio helped United gain control of the match and they created some chances to kill the game off as Garnacho brought a flying save from Maynard-Brewer, who also got down low to keep out Eriksen's effort.
United eventually got the second that made the game safe in the 90th minute as debutant Facundo Pellistri teed up Rashford and the in-form forward added a second after being played through by Casemiro.