The 14 placed League Two side must now prepare themselves for a glamour home tie against Premier League champions Manchester City at Rodney Parade.
The Exiles generally outplayed Tony Pulis' sluggish outfit and eventually heaved their way to a 2-0 victory thanks to a couple of second-half goals from Robbie Willmott and Padraig Amond, the latter being an absolute beauty...
They now face high-flying Leeds on Saturday in a match that could have a big bearing on promotion.
However, Konstantopoulos could not keep out Willmott's strike from outside the box two minutes into the second period and Amond fired a well-worked corner into the top corner 23 minutes from time to send Newport into the last 16 for the first time in 70 years. It was men against boys.
The club later tweeted: "Everyone connected with Newport County AFC would like to wish County's Number One Joe Day and his wife Lizzy all the very best!"
I said, 'Of course you can.
"We always knew there was a chance the birth would fall during the match, but Lizzie backed me and made it an easy decision for me to play the game", Day said. "But I wish I hadn't because they were absolutely breathtaking", he said, via Sky Sports News.
"He sprinted off the pitch and it's the quickest I've ever seen him move".
The Irish wing-back opened the scoring for Nuno Espirito Santo's men and then levelled just before half-time after goals from James Bolton and Josh Laurent had put the Shrews 2-1 in front.
Non-league Barnet's adventure in the competition came to an end as they lost 3-1 at fellow Londoners Brentford.