It took the visitors four minutes to open the scoring after a quick turnover in possession which the ball was fed to Joe Guest, who swivelled his body before slotting the ball low into the corner of Rhys Forster’s net.
The Rebels responded well and on eight ought to have been level.
Good work from Matty Lench down the right saw him drill the ball into the danger zone, where Sam Shaw was denied from point blank range by the Taunton stopper Jack Bycroft.
In what was a gritty affair, Luca Woodhouse met a Lench corner but was adjudged to have fouled before diverting his header just over the bar.
On 39 minutes Slough had a wonderful opportunity to get back into the game through returning striker Alfonso Tenconi.
The forward raced clear after a pinpoint pass from Woodhouse and got a strike off from the left of the area which went just wide and leaving the keeper stranded.
Just before the interval, Ben Long tested Bycroft from distance as a loose ball from a corner fell to him, but it went straight into the keeper’s arms.
Slough started the second half on the front foot. Aaron Kuhl whipped a ball from the left to the back post, Youssef Chentouf headed back across goal towards Tenconi who couldn’t make contact with the ball in a promising position.
Shortly after, Kuhl created another opening after finding Chentouf free on the right. He raced away but sliced his shot wide from the edge of the box.
Kuhl was dictating proceedings and tried a speculative effort for himself, but his effort rose high over the bar.
The visitors were under significant pressure, but ought to have doubled the lead on 60 minutes.
Nat Jarvis was found in acres of space in the box but somehow diverted his header wide from when the goal was at his mercy,
Once again, the Rebels responded well with Jackman volleying wide from a Lench cross.
Crosses and set plays were proving to cause Taunton problems, and on 67 they had Bycroft to thank for keeping them ahead.
Lench whipped in a corner from the left which found Woodhouse, who got a volley off in a crowded space, but the Taunton stopper superbly tipped the ball over the bar.
On 78, a well worked free kick from Lench and Kuhl ended with Tenconi having a headed effort which was blocked by a stubborn Taunton defence.
Slough were persistent on levelling proceedings in the latter stages; Lench again whipped in a ball which found Elliot Benyon, but he poked the ball wide at the near post.
The drama wasn’t done, and just before added time the visitors were reduced to 10 men.
Nat Jarvis lunged into a dangerous challenge on Rebels defender down the right-hand side and was shown his marching orders.
This spurred Slough on with 6 minutes added on.
Shane Bush met a Lench corner, with his header somehow being deflected wide as a goal looked destined.
Woodhouse was also so close to levelling deep into injury time, with his effort from the edge of the box being tipped around the post by the extraordinary Bycroft between the sticks.
Slough were unfortunate not to level and created enough chances to even be in the hat for the next round.
The Rebels are next in action when they head to Oxford City for a Boxing Day clash.
Taunton Town Lineup
Bycroft, Foulston, Grimes, Ball, Wannell, Chamberlain, James, Jarvis, Guest, Smith (Morgan 83), Sims-Burgess (Stearn 51). Unused substitutes: Budd, Irish.