Erik ten Hag's side made a strong start to the fourth-round match against League Two opposition, as goals from Bruno Fernandes and Kobbie Mainoo put them in charge inside 13 minutes.
But Newport responded admirably, halving the deficit before half-time through Bryn Morris and Will Evans then struck shortly after the break to equalise and spark real hope among the home fans of a monumental cup upset.
United restored their lead through, midway through the second-half, as Antony converted after Luke Shaw's strike had hit the post. Rasmus Hojlund put the match to bed late on, and the Premier League side will travel to face either Bristol City or Nottingham Forest in the fifth round.
It is a competition that represents United's only hope of finishing the season with some silverware, but the build-up to the match was largely dominated by talk of Marcus Rashford's absence, with the forward missing the match due to illness, having been absent from training on Friday just hours after reportedly attending a nightclub in Belfast.
There was no sign of United being distracted though, as they flew out of the blocks and offered a constant threat in wide areas.
The opening goal came just seven minutes into the tie, as Antony, potentially offside, was played in by Fernandes, cut it back to the United captain and the first-time finish was lashed into the back of the net.
Again Newport were cut open shortly after, this time down the right wing through Diogo Dalot, and he picked out Mainoo to sweep home his first goal for the club.
Alejandro Garnacho then smacked an effort against the bar, with Fernandes and Rasmus Hojlund waiting for a tap-in, as United threatened to run away with it and Newport showed no sign of being able to stem the tide.
But one moment of magic changed the complexion of the match, as the ball dropped for Morris 25 yards out and, with the help of a deflection off Lisandro Martinez, he found the far corner with a wonderful strike.
That left Newport right in the match heading into the second-half and within two minutes they were level. Adam Lewis played in a cross from the left, and Evans was sharpest at the near post to react and poke the ball beyond debutant Altay Bayindir.
Newport were firmly on top after that goal, looking well capable of forcing a replay or even snatching victory on home soil, but the next goal went United's way in a body blow for the hosts.
Luke Shaw cut onto his right foot, hit the post with a curling effort, and Antony was on hand to turn the ball into the empty net as the ball fell to him, his first United goal in 23 appearances this season.
The match remained in the balance heading into nine minutes of added-time, Newport summoning the strength for one final charge. It was cut short though, as Omari Forson's effort was saved and Hojlund pounced after a brief scramble, with Newport's race finally run.