Erik ten Hag's side led by two goals inside the opening 13 minutes at Rodney Parade, with Bruno Fernandes and Kobbie Mainoo on the scoresheet, but Newport fought back to level it at 2-2 courtesy of efforts from Bryn Morris and Will Evans.
Antony netted a third for the 20-time English champions in the 68th minute, though, before Rasmus Hojlund added another in stoppage time, and Man United will now take on either Bristol City or Nottingham Forest in the next round of the competition towards the end of February.
There was no Marcus Rashford for the Red Devils, with the forward missing through illness according to the club, but there was controversy surrounding his absence, as he was spotted at a Belfast nightclub on Thursday evening, which could lead to disciplinary action.
Head coach Ten Hag still named a very strong side, with Luke Shaw, Lisandro Martinez and Casemiro starting following recent injuries, while Altay Bayindir made his debut between the sticks, with Andre Onana not yet back from the Africa Cup of Nations.
The first-half chance of the match came for the 20-time English champions in the second minute, but Antony's curling effort was always wide of the Newport County goal.
Man United made the breakthrough in the seventh minute of the contest, and it was a super strike from Fernandes, who picked out the bottom corner from distance after meeting a cross from Antony, with the Brazilian finding himself free down the left.
Antony came close to scoring himself in the 11th minute after another brilliant move from the Red Devils, but his effort from inside the box was just too high.
Man United did double their lead in the 13th minute, though, with Diogo Dalot finding Mainoo, and the teenager swept his effort into the far corner with an excellent finish.
The visitors came so close to a third in the 18th minute when Alejandro Garnacho found himself inside the Newport County box, but the Argentina international's strike came off the crossbar.
Bayindir's first serious work in the Man United goal came in the 25th minute, with the goalkeeper making a smart save to keep out a strike from Newport's Evans.
Antony had another big chance in the 29th minute, with the Brazilian found by Garnacho inside the penalty box, but his effort was blocked by Ryan Delaney, before the forward had a strike tipped over the crossbar by Newport goalkeeper Nick Townsend.
The home side managed to halve the deficit in the 36th minute of the contest, and it was a quite incredible strike from Morris, who arrowed one into the back of the net off the post, albeit with the help of a deflection.
Newport had their best spell of the match in the period that followed, with Morris's effort certainly raising the crowd, and it was an uncomfortable end to the first half for Man United, but the Premier League side were able to enter the half-time break with a 2-1 lead; it was almost three in the final action of the second half, with Fernandes firing towards goal, only for Newport's goalkeeper to make a super stop.
Antony sent an acrobatic effort over the home team's goal early in the second half, with Man United on the front foot, but Newport incredibly levelled the scores in the 48th minute through Evans, who tapped a low cross from Adam Lewis into the bottom corner from close range.
Newport goalkeeper Townsend did well to deny Fernandes in the 54th minute, with Man United looking to hit back, before Bayindir had to keep out a header from Delaney down the other end of the field.
Man United came close to a third in the 61st minute through Martinez, but his close-range effort was deflected just wide of the post by his own teammate Hojlund.
Harry Charsley struck wide of the away team's goal in the 67th minute, but Man United regained the lead in the 68th minute, with Antony converting into the back of the net from close range for his first goal in nine months after Shaw had rattled one against the post from distance.
Garnacho was denied by Townsend in the 77th minute, before a medical emergency in the stadium led to the match being stopped, but it eventually got back underway at title before 6.15pm local time.
Harry Maguire made his return from injury as a late substitute, while there was also an appearance for Omari Forson, who is currently being heavily linked with a move away from Old Trafford.
Evans struck just wide of the post in the 90th minute, with Newport looking to make it 3-3 at Rodney Parade, and nine additional minutes were added.
The visitors made sure of their spot in the next round of the competition in the 94th minute, with Hojlund firing into the back of the net after Forson had been denied, with the Denmark international netting his eighth goal of the campaign.
Newport came close to a third late on through James Waite, but Bayindir made a smart save, before the resulting corner was hooked clear.
Man United's attention will now switch back to the Premier League, preparing to head to Wolverhampton Wanderers on Thursday, while Newport's League Two campaign continues against Swindon Town next weekend.