Slabber scored after 29 minutes of an impressive show by our youth team, who imposed themselves on the game from the off and were worthy of a greater scoreline.
Blackburn Rovers are the prize now and a first home tie of the competition will be welcome for the boys who have now triumphed at the Bescott Stadium, Walsall, the Reebok Stadium, Bolton and now the imposing Elland Road.
Shortly after the strains of 'we are Leeds' had finished echoing around Elland Road, Paul O'Donoghue found himself presented with a glorious chance to head us in front with only three minutes on the clock.
Dean Marney swung over a free-kick from the right and the centre-half found himself in space and thrust his header goalwards. Unfortunately, Leeds keeper Scott Carson was equal to the challenge and reacted with a spectacular save, leaving O'Donoghue to hold his head in disbelief.
Leeds then established a foothold in the game and Rob Burch raced smartly off his line to deny Craig Stiens.
Spurs play had a cohesive look to it - even without influential skipper Ben Bowditch. Ronnie Henry took the armband for the night.
Dean Marney slotted into the Bowditch role alongside Mark Hughes in the centre of midfield, with Lee Barnard and Nicky Wettner occupying the right and left berths respectively.
Wettner tormented his marker down the left during the opening stages, making several incisive runs. It was from one of those runs that the boys took the lead on 29 minutes.
He jinked down the flank with a combination of pace and trickery before firing over a waist-high cross that evaded the waiting Leeds defenders. Jamie Slabber, scorer of both our goals in the 4th round win at Bolton, was on hand at the far post to bundle the ball home and secure a deserved lead.
Marney was in the wars minutes later, coming off worse in a three-way collision involving Slabber and a Leeds defender. He carried on after treatment and, to add to his woes, was booked for a dubious body check as Leeds broke forward.
Leeds began the second half with a more positive air about them, not doubt a result of a half-time pep talk from coach Pop, the original Bryan 'Pop' Robson.
It was Pat Holland's side though, who came closest to next troubling the scorers. Hughes, Slabber and Barnard all took pot-shots at the Leeds goal during a frenetic goalmouth scramble. Matthew Kilgallon was the Leeds saviour with two brave blocks.
Hughes was next to go close following a Marney corner that drifted out towards the left of the Leeds penalty area. Hughes struck a stinging first time shot that swung and grazed the bar on its way over.
Despite Leeds gaining some territory for a short spell, it was Spurs that carried the goal threat - none more so than the dangerous Slabber. From the right of the area Slabber unleashed a vicious drive that Carson did well to block, but he surpassed himself from the rebound, pawing over our 17-year-old striker's second attempt.
Eleven minutes from time Leeds came tantilisingly close to a leveller. A cross swung over from the left cleared the back line and found the waiting Kinsella closing in on the back post. To the relief of the boys his shot flashed across goal with no Leeds player ready to reap the dividend.
Finally a home tie awaits in the quarter-finals, Blackburn, conquerors of Arsenal, are the next hurdle to be cleared by the boys in their Youth Cup campaign.
Team: Burch, Kelly, Herron, Hughes, Henry, O'Donoghue, Barnard, Marney, Sutton (Thomas 87), Slabber, Wettner. Subs: Rutherford, Galbraith, Thomas, McKie, Lee.
Leeds: Carson, Edwards, Keyes, McStay, Kilgallon, Keegan, Milner, Winter (Kinsella 66), Armstrong, Stiens (Shields 80), Woods (Mitchell 76). Subs: Coyles, Kreif.
By Richard Hubbard at Elland Road
Jamie Slabber scored the decisive goal that saw our Academy side through to the quarter-finals of the FA Youth Cup after a single goal victory over Leeds United.