Erik ten Hag's side have now won eight consecutive games in a row to move into the last four of the competition.
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United threatened to run riot when they broke the deadlock at Old Trafford thanks to Antony's brilliant curling strike from distance in the 21st minute.
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The Brazilian winger cut onto his left foot from 25 yards out before unleashing an unstoppable strike into the bottom left corner.
It was his fifth goal for the Red Devils since joining from Ajax last summer, with his goal reminiscent of his cracker away at Manchester City.
But they struggled after their opener, with Charlton having plenty of the ball in the second half where they were arguably the better team.
However the League One outfit couldn't convert any of their chances, with substitute Rashford grabbing United's second of the match in the 90th minute.
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The England international fired home from close range, before scoring another four minutes later.
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Rashford has now scored in six games in a row, with the United academy graduate on 15 strikes for the season.
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Seven of his goals have come after the World Cup break with the 25-year-old in a rich vein of form.
He has also found the back of the net in eight straight matches at Old Trafford - the first player to do so since Wayne Rooney in 2010.
The scoreline undoubtedly flatters United, but the result will give them plenty of confidence ahead of Saturday's Manchester derby at the Theatre of Dreams.