Newcastle escaped from Newport still in the Carabao Cup after scoring a late equaliser and then winning on penalties.
Brandon Cooper, who had a fantastic game for Newport, skied their sixth penalty over the bar to put the Magpies in the fifth round at the expense of the League Two side.
Earlier Joelinton saw his penalty saved by Nick Townsend before County substitute Ryan Taylor also saw his effort stopped by Mark Gillespie.
For most of the night it looked like Newcastle would not get to spot kicks and it took Jonjo Shelvey's superb curling effort three minutes from time to get them to a shoot out.
Newport celebrate taking the lead against Newcastle
Image credit: Getty Images
Prior to this, it seemed Gillespie's horrendous error would decide the contest after he allowed Abrahams' tame effort from 20 yards to bounce over him, after Scott Twine's dipping 30 yard shot had come back off the bar.
Steve Bruce would have rued that most of Newcastle's chances fell to Jacob Murphy but the Welsh side, who last year beat Leicester City in the FA Cup, were by no means outclassed by their Premier League opponents and could have added to their score if Padraig Amond or Scott Bennett had hit the target with decent chances.
How do Newcastle solve their scoring problems? Their opening Premier League match against West Ham seems a long time ago. That 2-0 victory with plenty of other chances suggested it would be a much-easier watch for the Toon Army this season. Since then, they managed only one shot at goal in the following two league matches - the much-debated penalty against Tottenham.
On this occasion they had 27 shots on goal against Newport but most were long-range efforts - such as Shelvey's stunning strike - or snuffed out at source by, admittedly, superb challenges from Brian Cooper and Mickey Demetriou in particular - and there was little clever or deft in their play in the final third.
Callum Wilson looked sharp when he came on, as you would expect against League Two opposition, but they patently lack a creative force with Allan Saint-Maximin out the side and you feel more help is needed in this area if the Magpies are to avoid a relegation scrap this season.
Man of the match: Nick Townsend (Newport)
If you are to cause a cup upset you need your keeper to have a decent game - and Townsend certainly did.
There were not necessarily highlight or camera saves but either coming out and blocking shots or parrying away long-range attempts to safety; and it seemed his efforts would be rewarded with victory. Even in the shoot-out it looked like he could be the hero when he stopped Joelinton's kick and there can be little blame attributed to him for Shelvey's superb equaliser.
Newport: Townsend 8*, Haynes 7, Cooper 8, Dolan 7, Shephard 7, Sheehan 6, Bennett 7, Twine 7, Demetriou 7, Abrahams 7, Amond 6,
Subs: Taylor 5, Wilmott 6, Collins 6.
Newcastle: Gillespie 5, Manquillo 6, Krafth 6, Fernandez 6, Lewis 6, Shelvey 7, S Longstaff 6, Murphy 7, Almiron 5, Fraser 6, Carroll 5.
Subs: Joelinton 5, Wilson 7, Schar 6.
5' GOAL! Abrahams hits home and it is thanks to terrible goalkeeping from Gillespie who let his fairly tame 20-yard bouncing effort over him. It came after a superb dipping thirty yard effort from Twine came back off the bar.
87' GOAL! It looked like it wouldn't come but it did in style from Jonjo Shelvey. He twisted and turned leaving Wilmott on the turf before curling a superb effort into the stanchion in the far right corner.
Newcastle penalty 2 - Joelinton misses. Saved by Townsend.
Newport penalty 4 - Taylor misses. Saved by Gillespie. A tame effort.
Newport penalty 6 - Cooper misses. He skies the ball high over the bar.