Harry Jessop bagged a brace before the visitors got one back midway through the second-half.
Jessop went close inside two minutes, volleying wide of the post from the edge of the area.
Seven minutes later, the Iron took the lead. Winning a corner after some good work between Levi Gallimore and Cameron Wilson, the set piece was played out to Jessop who curled in the opener from outside the box.
Raynner Silva looked to double the advantage, drawing a save from goalkeeper James Anderson after trying his luck from distance.
In the 22nd minute, Jessop and Oliver Lobley combined, with the latter latching on to Jessop’s ball down the left. The defender picked out Levi Gallimore but he missed the target.
On the half hour mark, Levi Gallimore went close again after being set up by his namesake Dan Gallimore. The follow-up fell to Joey Dawson whose effort from 30 yards out flashed wide.
However, it was Jessop who added his and the Iron’s second, with 38 minutes on the watch. The forward found himself on the end of Harry Baker’s delivery and netted from close range to make it 2-0.
There was almost a third, minutes later. Silva’s effort from range was deflected behind by a Morpeth player.
The visitors started the second period brightly. Baker made an important defensive header to clear a cross away before Dan Gallimore got his body in the way of another incoming shot.
Five minutes into the half, Levi Gallimore rifled an effort towards goal but it was straight at the Morpeth shot-stopper who was equal to it.
The visitors halved the deficit in the 59th minute as substitute Hakmoaui made it 2-1 from close range.
Morpeth should have been level after spurning a golden opportunity in the 68th minute. The visitors were awarded a free-kick which was denied by the wall before Tom Collins made the save from the rebound.
The ‘keeper was called into action moments later to deny Malone again, diving off to his right to keep the Morpeth player out.
Opposition goalie Anderson made a fine double save with 15 minutes remaining. Dawson made inroads with Jessop who picked out Silva’s run but the ‘keeper was on hand to stop both efforts.
Charlie Barks then made a goal-saving block from Jackson Ward late in the day to preserve the lead.
Jessop almost completed his hat-trick after doing all the hard work, getting in behind and rounding the ‘keeper, only to be thwarted by a challenge from a Morpeth defender.
The striker was forced to withdraw from the game in the final five minutes, going down after the tackle.
In added time, Collins saved Milburn’s header to keep the visitors from equalising.
The ‘keeper made an even more impressive stop, two minutes later. He single-handedly denied a Morpeth player before Barks cleaned up at the back.
The Iron held on for the win, with Collins making one final save to keep out Ramsey’s low effort at 2-1, and will go on to face West Bromwich Albion in Round Three.
IRON U18s: Collins, Baker, Lobley, Dawson, Barks, Shrimpton, D Gallimore, Wilson, L Gallimore (Wilkinson, 60), Jessop (Virgo, 86), Silva
IRON U18s UNUSED SUBS: Chadli, Train.
MORPETH TOWN U18s: Anderson, McHarg, Hutchinson (McKeown, 45), Foreman, Fawcett, Milburn, Paul (Hakmoaui, 45), Walker, Malone (Fox, 85), Ward, Ramsey.
MORPETH TOWN UNUSED SUBS: Dippenaar, Crowle.