NewPort Goalkeeper Joe Day cut short his team's celebrations to race to his wife's side as she gave birth during the match.
Day had just kept a clean sheet as Newport won the FA Cup fourth round replay 2-0.
The win set up a FA Cup fifth round tie with Manchester City for the League Two side.
"We'd had no contact since before the game, but Lizzie told me there were two little girls and they were flawless and waiting for me to arrive at the hospital - and we've been here ever since".
"The professional that he is, he turned his phone off", Newport manager Michael Flynn told BT Sport after the game.
Boro were well below par against the League Two outfit as they crashed out in the fourth round replay on Tuesday night.
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"I didn't know the girls had been born when I ran off the pitch at the end".
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's side level at the break.
Newport, who are currently 14th in League Two, will now face Pep Guardiola's Manchester City at home on February 16 for a place in the quarter-finals.
Matt Doherty's second-minute goal opened the scoring but goals from James Bolton and Josh Laurent put Shrewsbury in front in the tie once more.
"We are going to have to be prepared - I might play nine at the back".