Leeds, sixth in the second tier Championship and 53 places higher up the ladder, went ahead through Gaetano Berardi's ninth-minute strike and appeared to be heading through until Conor Shaughnessy's own goal 15 minutes from fulltime.
After McCoulsky's headed winner, Leeds's day went from bad to worse as they had Samuel Saiz sent off in stoppage time.
"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," Newport manager Michael Flynn said.
"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."
Leeds made nine changes to their lineup but took the lead when Berardi's 30-metre low shot took a crucial deflection off his team mate Pierre-Michel Lasogga to leave Newport keeper Joe Day wrong-footed.
Newport pushed for an equaliser but needed a slice of luck to draw level when Frank Nouble's cross deflected off of team mate Padraig Amond and then Shaughnessy on its way onto the net.
The best was yet to come though for the hosts though as McCoulsky, on as a 69th-minute substitute, rammed in a late header from a corner.
"It is a brilliant feeling. The gaffer told me to come on and affect the game as much as I can," he said.
"I can't remember most of it, it was a blur to be honest. It is definitely one of my best goals, to knock Leeds out of the FA Cup is what dreams are made of."
It is the second season in succession Leeds have suffered a humiliating exit in the Cup after losing to minor league Sutton United in the fourth round last time.
(Reporting by Martyn Herman; Editing by Toby Davis)
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