The Foxes, who impressively beat Grimsby Town, Sunderland, Doncaster Rovers and Tranmere Rovers to reach the last-eight headed to south Wales after the original tie was postponed last month.
Despite City showing encouraging signs in the early exchanges, Michael Flynn’s home side went in front on 13 minutes as Scot Bennett capitalised on a loose ball in the area to slam it into the net.
Elsewhere, Viktor Johansson made a brilliant fingertip save to keep out Matty Dolan’s free-kick before the half-hour mark, while Bennett missed out on another chance, nodding it just wide.
Vontae Daley-Campbell so nearly equalised for Leicester soon after as he burst into the 18-yard box and arrowed a powerful effort narrowly wide of Nick Townsend’s post.
Johansson's quality was required again at the other end in the final moments of the first half, though, as the young Swede got a hand on Josh Sheehan’s low effort to keep it 1-0.
After the interval, with Khanya Leshabela on in the place of O’Connor, giving the Foxes an extra man in midfield, the visitors posed an heightened threat to Newport’s advantage.
On 50 minutes, Daley-Campbell’s cross found Ali Reghba in the area and, although the Irishman scuffed his effort, it came out to Luke Thomas to hit it at goal first time.
Only a remarkable save from Townsend, tipping it over the bar, denied the England youth international from getting on the scoresheet and it seemed to galvanise City’s youngsters.
George Nurse was unlucky to drag a low drive wide for the Exiles, but there was a noticeable impetus from the away side in chilly conditions in south Wales.
With 16 minutes to play, Reghba showed his physical attributes to win it from Kyle Howkins before cutting inside and lashing a shot inches over the bar as Leicester continued to threaten.
Another excellent opportunity came City's way with three minutes of normal time remaining, with the ball coming through to Hirst in the area. The Development Squad's top scorer, though, sent it wide.
In the final seconds of four added minutes, Newport advanced on the counter as substitute Jordan Green located Padraig Amond in the area, but Johansson was there to save it comfortably.
The smallest of margins stood between Luke Thomas and an equaliser.
Perhaps City's best chance to claw themselves back into the tie came five minutes after the half-time break, but Ali Reghba initially appeared to have missed out on the opportunity to draw level.
However, it landed perfectly for Luke Thomas to go for goal on the edge of Newport's penalty area. The England youth international hit it sweetly, but there was Nick Townsend to tip it over the bar.
Defender Luke Thomas applies pressure on Newport shot-stopper Nick Townsend.
A force in attacking positions and defensively sound, Luke Thomas continues to make a name for himself in the Club's Development Squad and Tuesday night was no different.
If not for an excellent save early on in the second half, the 18-year-old could have equalised for the Foxes, but his overall contribution was still key for the visitors.
Where does it leave us?
The match became a real physical battle in the second period and City withstood the challenge.
Newport will host Salford City in the Leasing.com Trophy Semi-Finals on Wednesday 19 February for a place in the final against Portsmouth or Exeter City at Wembley later this year.
Leicester's showing in the 2019/20 campaign, meanwhile, is their best-ever since electing to enter an Under-21s side into the competition back in 2016 after previously only reaching Round 3.
Calvin Ughelumba rises to meet a Foxes corner in south Wales.
City’s youngsters will benefit from a 10-day break from competitive action, when Tottenham Hotspur visit Holmes Park in Premier League 2 on Friday 14 January (7pm GMT kick-off).
City: Johansson; Daley-Campbell (Pennant 79'), Johnson (c), O'Connor (Leshabela 45’), Ughelumba, Thomas; Wright, James; Eppiah, Hirst, Reghba
Subs not used: Stolarczyk, Kranthove, Tee
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