But they were forced to fight back from a goal down after the highly-rated Reds' youngsters had taken a deserved lead midway through the first-half.
Lively striker Adam Orme had already gone close on several occasions, but made no mistake when seizing on a poor back-pass from Connell Rawlinson.
The hosts levelled matters eight minutes before the break with a set-piece move which saw Barry Haralambous head in at the far post from a corner.
A much better second-half performance from TNS started with a goal two minutes after the interval, Charlie Proctor scoring at the third attempt after Llanelli keeper James Vyse had blocked his first two efforts.
And it got even better for the home side on 68 minutes, when the visitors' defence was caught in possession by Marcus Giglio, who provided for Ash Jones to race on and score from just inside the edge of the box.
Despite looking comfortable on their lead, the Saints engineered a tense finish with some slack defending allowing an unmarked Orme to add his second from a Henry Hughes left wing cross.
But they held out for the last eight minutes, despite Hughes and Jordan Langley going close to an equaliser, to set-up a last four meeting with Swansea outfit West End.
Cardiff meet Wrexham in the other semi-final with ties to be played on a neutral ground on or before Sunday March 27.
Photo: Marcus Giglio and Ash Jones celebrate with goal scorer Barry Haralambous.