The youngsters became the first Luton side to reach this stage of the competition since 2006-07, with their feat ever more remarkable by the fact they had to play with 10 men for an hour.
However, the visitors held firm to book a tie at either AFC Bournemouth of Oldham, while stretching their superb unbeaten run to 22 games.
Boss Paul Driver kept faith with the same starting line-up that had beaten Colchester and Leyton Orient in the previous two rounds.
Luton began well too as Kavan Cotter’s low show found Freddie Hinds, whose instinctive shot was well saved by Joe Searson-Smithard.
Hatters were in front after 15 minutes though as Alex Atkinson’s defence-splitting pass found Hinds who beat the onrushing keeper to score his 13th goal of the season.
The second then arrived five minutes later as Jack Snelus played a one-two with George Murray and drove into the area before producing an unerring finish.
Driver’s side threatened a third as Kyran McGhan-George’s header was saved, while Cotter and James Justin shot wide.
However the game turned on 29 minutes when Magpies striker Montel Gibson raced through on goal and was brought down by Frankie Musonda just inside the penalty area.
The Town skipper, who recently signed his first pro deal at Kenilworth Road, was dismissed and Gibson converted the resultant spot-kick.
Despite their numerical disadvantage, Luton went close to a third as Hinds was inches away from forcing the ball home, while Snelus, Hinds and Murray all went close.
After the break, any fears of a County comeback evaporated just five minutes into the second half as Murray rose at the back-post to nod home via the crossbar.
Luton spurned a glorious chance to make it 4-1 on 67 minutes when Cameron McJannet’s left-wing cross was missed by Murray from six yards.
Snelus almost capped the evening with a superb strike only to see his rasping 25-yard drive tipped over the bar.
Hatters: Gooch, Justin, McJannet (Bean 68), Famewo, Musonda, McGhan-George, Cotter, Atkinson, Snelus, Murray (Bakinson 73), Hinds.
Subs not used: Verney, Brown, Shamola.