Erik Ten Hag's side took an early 2-0 lead through Bruno Fernandes and a first Man United goal for Kobbie Mainoo.
Newport then struck back to level things up with the side who are 75 places above them in the football league pyramid, with goals from Bryn Morris and Will Evans either side of half-time.
However Man United's quality showed when Antony fired home after a Luke Shaw curler cannoned back off the post, before Rasmus Hojlund wrapped things up in stoppage time.
The first competitive game between the two clubs had been billed as the biggest in Newport's history, with just over 9,000 crammed into modest surrounds for a tie expected to earn the League Two club around £400,000.
But County's recent cup pedigree – wins over Leicester and Leeds, as well as holding Tottenham to a Rodney Parade draw in 2018 – and a seven-game unbeaten run put United on notice they could not expect a smooth ride by the banks of the River Usk.
Ten Hag could almost feel the breath of home supporters standing just yards behind him and there was little space on the pitch either during a frantic opening.
There was no Marcus Rashford in the United squad with the England striker having reported ill on Friday after reportedly spending the previous evening at a Belfast nightclub.
Ten Hag said before kick-off that Rashford's absence was an "internal matter" and that he would "deal with it".
Turkey goalkeeper Altay Bayindir came in for his debut with regular number one Andre Onana at the Africa Cup of Nations.
But United were stronger – at least on paper – for some time with Casemiro, Lisandro Martinez and Luke Shaw back in the starting line-up, while Harry Maguire was on the bench after suffering a groin injury on December 12.
Newport named the same side which had accounted for Eastleigh in round three with the presence of former farmer Evans, their United-supporting 18-goal marksman, presenting the kind of charming storyline the FA Cup tends to supply.
Fernandes did his best to remove the romance after seven minutes when receiving a ball from Shaw and finding Antony.
The Brazilian played an instant return and Fernandes drilled a shot of rare purity past Nick Townsend within the blink of an eye.
County fans, raucous at kick-off, were silenced and left despairing again as Diogo Dalot motored down the right to set up 18-year-old Mainoo, who carefully caressed his finish into the bottom corner.
Alejandro Garnacho might have made it worse but rattled the crossbar and Antony drifted in from the right flank to force Townsend to tip over.
Newport kept Bayindir honest with efforts from Evans and Ryan Delaney, but Morris' 36th-minute thunderbolt really came out of the blue.
Morris, at Grimsby when the Mariners reached the quarter-finals of the FA Cup last season, chested the ball down and unleashed from 25 yards to beat Bayindir with the help of a deflection off Martinez.
Townsend denied Fernandes with the final kick of the first half and the importance of that save was underlined within two minutes of the restart.
Adam Lewis crossed and Evans stole in front of Raphael Varane to level with an outstretched foot.
With it becoming increasingly difficult to distinguish who was the Premier League side, and Ten Hag being reminded of his employment prospects in the morning, Bayindir held Delaney's header under his own bar.
Harry Charsley drove wide, but United re-took the lead after 68 minutes as Shaw curled on to a post and Antony slotted the rebound.
Maguire was sent on late to preserve United's slender lead before Hojlund extended his scoring streak by pouncing on a loose ball and drilling home.
