They scored three of them, the first with the game only three minutes old, and eventually eased to victory in the second half after Paul McCallum had equalised for the hosts with a fabulous strike just after half-time.
James Clarke restored Newport's lead on the hour before Will Evans sealed the win in the 79th minute, collecting the ball on the edge of the area after three players had combined to earn space down the left, beating his marker with his first touch and arrowing a left-footed shot into the far corner. It took an excellent save from Joe McDonnell three minutes later to stop Bryn Morris giving the scoreline what would have been an unfairly lopsided look.
In the buildup Richard Hill, the Eastleigh manager, described this as "the biggest game in the club's history without a shadow of a doubt".
They had played in the third round before, against Bolton in 2016 and Brentford the following year, but never had the opportunity, at home and against relatively lowly opponents - Newport sit 17th in League Two - to land one of the non-league game's great prizes: a home cup tie against one of the best-supported clubs in world football.
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