All four goals of the first round tie were netted in the final 45 minutes, with Flaherty's acute finish breaking the deadlock two minutes after the restart at Whitley Park before the Lions, who compete in Northern League Division One, quickly levelled courtesy of Connor Walker's close-range tap-in.
The young Magpies, however, regained their advantage through Flaherty's penalty, following a handball inside the penalty area, before a trialist's composed finish in the closing stages sealed the hosts' passage into the last eight.
Flaherty, making his first appearance of the 2021/22 campaign at under-23 level, told nufc.co.uk: "It was a tough test tonight. It was different to what we were used to but we all adapted well to it and the boys came through in the end.
"It was a bit sloppy in the first-half and I think we gave them more of a chance than they should've had but, in the second-half, we dug deep, pulled through and got the goals we needed. We're buzzing to go through.
"In the first ten minutes, especially, they were quick to their challenges and left a bit on you so I think it took us a little while to adapt to the way they played. Once we did that, we handled it well and showed that in the second-half.
"To get the one-nil lead set a good precedent for the game. We shouldn't have let them back in as quickly but, after we conceded, we handled it well on a whole.
"The second and third goal we scored was really important as it gave us that cushion. If we were 2-1 up, they could have easily got back into it but the third was key, a very good finish."
United's second-string have won the competition a record 38 times, last winning the trophy back in 2017/18 following a 3-1 triumph over Morpeth Town at St. James' Park.
Flaherty admitted the squad are determined to progress further in the tournament in a bid to end the season with a piece of silverware.
The 19-year-old winger added: "We want to try and win every competition we take part in. "We're in a few other cups and I think this is a one where we have a really good chance to go and express ourselves on a bigger stage in terms of facing senior sides.
"The lads are looking forward to the rest of it and hopefully we can go a long way."
