McCoulsky's winner in the 89th minute was deserved reward for the home side, who were superb throughout.
Conor Shaughnessy's own goal drew them level with 15 minutes remaining after Gaetano Berardi's deflected ninth-minute opener for the visitors.
Leeds' misery was completed when Samuel Saiz was sent off in time added on.
Newport's dramatic recovery caps an incredible few months for the Exiles, who only stayed in the Football League last season with a winner in the closing stages of their final match of the campaign.
Their last-gasp heroics continued against Leeds in a win that will be remembered in south Wales for years.
The joy of 53
The omens were there for Newport before the tie.
They were separated by 53 places in the Football League, the same margin between Coventry City and Stoke City before the Sky Blues' surprise win on Saturday.
Fifty-three places also separated Cardiff City and Leeds in 2002, when the Welsh side stunned the David O'Leary team that was contending for the Premier League title.
Newport and Leeds had not met in the FA Cup since January 1949, when the Welsh side won 3-1 at Elland Road in a third-round tie, and this was again to be a famous afternoon for County.
The two teams were however hardly strangers, having already met this season at Elland Road in the Carabao Cup, with Leeds winning 5-1.
Newport nevertheless created several chances in the first half of that game and did so again at a freezing Rodney Parade.
First, Padraig Amond spun and shot after Leeds failed to clear a corner, but Mateusz Klich blocked on the line.
Skipper Joss Labadie then saw two efforts blocked as Newport bombarded Leeds with crosses, before Amond and Robbie Willmott fired over from tight angles.
County trailed at the break, but were unfortunate to do so.
Nine changes cost Leeds
Leeds looked vulnerable and lacked fluency, which was perhaps no surprise given that only Berardi and Kalvin Phillips maintained their places from the side that started the goalless draw with Nottingham Forest in their last game.
Thomas Christiansen's side were on the back foot for much of the match yet took the lead when Berardi's 30-yard drive struck Pierre-Michel Lasogga's leg and wrong-footed Newport goalkeeper Joe Day.
The goal sparked wild celebrations and flares from the travelling fans, but Newport were unperturbed and missed a handful of good chances to level before and after the break.
Amond volleyed wide acrobatically just before the hour mark, while a wicked delivery from Scot Bennett just needed a touch as it fizzed across Andy Lonergan's goal.
Leeds, who tried to slow down the game after the break, occasionally threatened on the counter, with Lasogga forcing Day to parry a robust shot.
Labadie volleyed over with 20 minutes remaining before the home side's magic moment arrived with Frank Nouble's cross deflecting off Amond and then Shaughnessy with Lonergan helpless in Leeds' goal.
Amond's late header from another Willmott set piece was held by the goalkeeper and substitute Paul Hayes fired over on his debut as the Exiles looked like they would just fall just short of a famous win.
However, substitute McCoulsky struck and County held on comfortably once Saiz saw red for his part in a melee.
Man of the match: Robbie Willmott
What they said:
Newport County manager Michael Flynn, speaking to BBC One Wales: "This is much better, beating Championship teams like Leeds. I don't want to be winning relegation battles, I want to be winning FA Cup ties and climbing the table.
"I thought we were the better team in the first half. The goal was unfortunate, but we had chances and kept going. I told them we don't have to win the game in the first 10 minutes of the second half and we got crosses into the box.
"It was an outstanding header. He (McCoulsky) has 11 goals for us this season and he has been superb for us. He was hungry and wanted it and I am delighted."
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