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Substitute Rashford produced two fine finishes to round off United's victory, after Antony's stunning strike broke League One side Charlton's resistance in the first half at Old Trafford on Wednesday (AEDT).
Charlton – the only non-Premier League team in the last eight – produced a stubborn performance to stay in the contest until the closing stages, but lacked the quality to trouble its illustrious opponent.
Erik ten Hag's men are now potentially three games from ending a six-year trophy drought after reaching their third Carabao Cup semi-final in four seasons.
Alejandro Garnacho was heavily involved as a much-changed United team started well, teeing up Diogo Dalot for a wild left-footed strike before going close from long range.
It was Antony who opened the scoring 21 minutes in, curling a sublime left-footed strike beyond a diving Ashley Maynard-Brewer and into the top-left corner.
Fred almost doubled United's lead in spectacular fashion when his 25-yard free-kick rattled the inside of the post, ensuring Charlton were just one goal down at the break.
Anthony Elanga was denied by the offside flag when he converted from close range after the restart, before Maynard-Brewer kept Charlton in the game with a fine stop from Garnacho's bending right-footed attempt.
United finally made the victory safe as the game entered stoppage time, the in-form Rashford beating Maynard-Brewer with a chipped finish after taking Facundo Pellistri's pass in his stride.
There was still time for the England forward to find the back of the net again, rounding the goalkeeper before hammering a left-footed finish high into the net.