Aaron Wildig's goal gave County a third-minute lead at the Silverlake Stadium, but the non-league hosts hit back at the beginning of the second half, through Paul McCallum.
James Clarke restored Newport's advantage with an hour gone and leading scorer Will Evans struck a third towards the end to confirm the meeting with Erik ten Hag's side, in what will be our first fixture after the winter break.
The match at Rodney Parade will kick off at 16:30 GMT on Sunday 28 January and it is being broadcast live on BBC One and BBC iPlayer in the UK.