The boys looked comfortable winners after an hour with Slabber's double on the scoreboard, but a 78th minute reply and a last minute penalty almost changed the direction of the tie.
Keeper Rob Burch was the late hero with a spectacular penalty save and the boys now head to Leeds United in round five.
Both sides struggled to establish a pattern in the opening stages of the game. Burch had to be alert and brave to make a block at his near post, while Slabber forced a smart save from the Bolton keeper.
Otherwise, the first 15 minutes amounted to cagey sparring between teams relatively unfamiliar with each other.
While the atmosphere of 1,500 eager youngsters would not equate to the high octane decibel levels of the Lane on the night, the futuristic Reebok Stadium is a magnificent arena.
The crowd did make themselves heard after 20 minutes. A cross from the right caused major confusion in the Bolton penalty area and Slabber was on hand to drive the ball home to a chorus of boos amid Spurs delight.
Pat Holland's boys pretty much bossed the remainder of the half, any Bolton threat was dealt with admirably by Burch - whose handling from crosses was superb.
Dean Marney was carrying the threat from midfield, ably supported by the lively Nicky Wettner down the left flank.
Marney attempted a pot-shot from 30 yards that had the keeper frantically back peddling, while a Wettner effort on goal from distance was deflected narrowly wide. From the resulting low driven corner John Sutton was unlucky with a header that drifted just wide of the keeper's left hand post.
Wettner was then booked for an enthusiastic challenge, one of many stoppages forced by a slightly fussy referee.
Minutes before the break a Marney free-kick was heading for the corner before an outstretched boot sent his effort off course and Slabber nearly troubled the scorers again, but the keeper did well to block his shot.
The probing was well and truly over at the beginning of the second half as both sides cut loose and went for the attacking option. Wettner was offered a clear shooting chance just after the interval, but screwed his shot wide.
He made amends by firing over a superbly timed cross minutes later that found Sutton who was foiled.
The boys were then on the defensive for a shot spell, a brilliant last-ditch block by Paul O'Donoghue from a low cross denied Bolton a leveller and a 35 yard volley from the No.4 flashed past Burch's post.
It was the boy Slabber again who doubled our advantage on the hour. Bowditch fed the prolific first year trainee to the right of the Bolton penalty area and Slabber jinked past his man and accelerated before firing a shot low and inside the keeper's right hand post - busting the net in the process.
Bolton reduced the deficit on 78 minutes when their No.11 came out better from a block tackle on the edge of the area. He found himself in the clear, reacted smartly and kept his composure, and slid the ball beyond Burch and into the net.
Bolton were fired by a sense of urgency and wasted little time on celebrations. But it was Bowditch who went close next with a high lofted shot that fell just wide.
Slabber then very nearly completed his hat-trick. A deep corner by Marney was met by Henry and his header back across goal was met by the 17-year-old. His header hit the underside of the bar and somehow Bolton scrambled clear.
Lee Barnard returned from injury and replaced Marney with four minutes remaining - four nervous minutes away from a trip to Elland Road to face Leeds United in round five.
The nerves were stretched even further right at the death when the referee awarded a penalty for handball against Henry. Protests fell on deaf ears and it was down to Burch to save the night. And save it he did with a fine save from the Bolton No.6 high to his left.
Mission accomplished. Next stop Elland Road.
Team: Burch, Kelly, Henry, O'Donoghue, Herron, Bowditch, Hughes, Marney (Barnard 86), Wettner, Sutton, Slabber. Subs: Rutherford, McKie, Lee, Galbraith.
By Richard Hubbard at the Reebok Stadium
Two goals from Jamie Slabber saw our Academy side progress to the fifth round of the FA Youth Cup after a dramatic 2-1 win over Bolton at the Reebok Stadium.