Padraig Amond, who scored the winning goal midway through the second half at Rodney Parade, told BBC Radio Wales: "They've had two healthy baby girls so congratulations to Joe and his partner Lizzie and the two little girls as well".
"It's a free shot for us and I'm sure our players will surprise them".
Seeing off 'Boro means that Newport have earned themselves a home tie in the fifth round and will therefore be welcoming Man City to their place on 16th February.More news: London Zoo Sumatran tiger Melati killed in fight
Boro were well below par against the League Two outfit as they crashed out in the fourth round replay on Tuesday night.
A penalty appeal for handball by Middlesbrough midfielder Adam Clayton was waved away and 40-year-old Greek goalkeeper Dimitrios Konstantopoulos made a string of saves to keep Tony Pulis' side level at halftime.More news: Tata Motors stocks plunge most in 26 years after Q3 results
"Nothing was really happening at midday on Tuesday, but as I was driving to the game Lizzie called me to say her waters had gone". I said 'Of course you can, do you want a lift?'
He sprinted off, it's the quickest I've seen him move, but all our thoughts are with Joe and Lizzy and the arrival of the twins.
Ivan Cavaleiro then completed the Wolves comeback in the second half.More news: Kenneth Lilly Charged With Attempted Murder In I-35W Bus Driver Shooting
Non-league Barnet's adventure in the competition came to an end as they lost 3-1 at fellow Londoners Brentford.