It was no less than the lads deserved after dominating for 88 of the 90 minutes as only some dogged defending from the Wiltshire outfit kept them at bay.
But it took until the 85th minute to break the deadlock when Slabber flicked home Owen Price's corner at the near post.
It always appeared to be a case of 'when' rather than 'if' we would score with Mark Yeates and Lee Barnett continually probing away in midfield, Nicky Wettner and Phil Ifil always willing to join in down the flanks and Slabber and the lively Michael Malcolm ready to pounce on any chances.
At the other end, the defensive pair of David Tyrie and Danny Foster were only breached once and Paul Rutherford was ready for the one vital save that came his way.
We started brightly and barely a minute had passed when what turned out to be the best chance of the half arrived.Malcolm showed a clean pair of heels down the left and hit the perfect cross for the arriving Barnett but the midfielder careful side-foot fizzed a yard wide.
Barnett was there again to volley a difficult ball over after 10 minutes and three minutes later a fine move down the right ended with Slabber looping a header towards goal that Steve Smith dealt with comfortably.
The teams cancelled each other out for the next 25 minutes until Tyrie lashed a tricky chance over and Wettner was foiled by a clearance off the line just before the break when Smith spilled Tyrie's long throw and the wing-back pounced onto the loose ball.
The second half was one-way traffic as Barnett and Yeates' influence grew in midfield and the lads used the full width of the pitch with Ifil joining in to good effect down the right.
He provided a cross that caused havoc after our best move of the game after 53 minutes and was there again to link up with Yeates whose cross was headed inches wide of his own goal by Luke Garrard.
Typically, Swindon then broke away to carve out the best chance of the game, their first, as Jamie Cheeseman found Mark Draycott unmarked only for the striker to miss the target from 10 yards.
That was after 71 minutes and a minute later, Rutherford acrobatically tipped away Draycott's lofted poke towards goal as the visitors enjoyed their best spell.
It wasn't long before play switched back to the other end as Dave Bampton somehow cleared after Smith made a hash of a clearance but the keeper was back between the posts to save Slabber's follow up.
Tyrie fired over from 20 yards and you just sensed a goal was coming - it finally arrived with five minutes remaining.
Substitute Price swung in a near post corner and Slabber rose highest to plant his header into the top right corner.
1min: Early Malcolm break down the left, low cross met by Barnett who side-footed a yard wide.
10mins: Loose ball in box fell to Barnett who volleyed difficult chance over from 10 yards.
13mins: Decent move down the right ended with Slabber's looping header from Ifil's cross but no real problems for Smith.
39mins: Smith spilled Black's corner but Tyrie couldn't keep high-bouncing ball on target.
42mins: OFF LINE. Smith again problems to deal with ball in, this time lost Tyrie's long throw under pressure from Malcolm, Wettner onto loose ball but blocked on the line by Garrard.
45mins: Black flick header from Tyrie cross deflected just wide.
Half-time: Spurs 0-0 Swindon
53mins: Best football of the game so far ended with wicked low Ifil cross scrambled away to the edge of the area where Yeates volleyed straight at Smith.
58mins: SUB. McKay for Hambridge.
59mins: Deep Tyrie cross picked up by Wettner left-hand side, low ball back in missed by Malcolm.
63mins: CLOSE. Tricks and a great cross from Yeates matched by headed clearance from Garrard under pressure from Malcolm that brushed the woodwork.
71mins: CHANCE. After 25 minutes of one-way traffic, Swindon broke and created the best chance so far. Cheeseman broke down the left and his cross found Draycott unmarked, 10 yards out, but he fired wide.
72mins: SAVE. Rutherford at full length to tip away Draycott's poke towards goal.
73mins: Bampton produced miracles to deny Malcolm and Wettner after Smith miscued, Slabber onto the loose ball but Smith back in time to save.
79mins: SUB. Price for Black.
80mins: Foster broke forward and fired just over from 20 yards.
85mins: GOAL. Slabber finally broke the deadlock, heading home Price's near post corner. 1-0.
90mins: SUB. Bullman for Pook.
Line-ups: Spurs: Rutherford, Ifil, Wettner, Tyrie, Foster, Dobson, Barnett, Yeates, Slabber, Malcolm, Black (Price, 79). Unused subs: Noto, Eyre, Perry, Francis.
Swindon: Smith, Taylor, Oliver, Bampton, Garrard, Taylor, Hambridge (McKay, 58), Pook (Bullman, 90), Daycott, Cheeseman, Stroud. Unused subs: Lapham, Salter, Jones.
Jamie Slabber arrived in the nick of time to head our Academy side into the fifth round of the FA Youth Cup on Wednesday night.