Yashwa Romeo struck twice in the first half to give the Rebels a 3-2 half time lead, but Thatcham struck back after the break and hit the winner after James Pritchard had levelled with just three minutes remaining to pile on the misery for Slough.
Slough were able to welcome back Eamonn McGonigley, Ryan Fenton and Jamie Jackson back to the team as the Rebels looked to stem the flow of goals conceded.
Eager to put the midweek display against Windsor & Eton behind them, Slough went on the attack with Platt firing just wide from 12 yards. But just four minutes had elapsed before Thatcham took the lead. A header down was struck across Jackson and into the back of the net by Jemel Johnson from six yards out.
Minutes later, Slough had a great opportunity to equalise when Jordan was played in down the right hand side. His cross to the back post was met powerfully by McGonigley, who headed the ball downwards and forced a good save from Strudley before the ball was cleared. On 12 minutes though, Slough broke through and grabbed an equalizer courtesy of Yashwa Romeo, who seized upon a mistake from Sam Rea and slotted the ball into the bottom corner.
Just two minutes later and Slough fans thought their luck had changed when Danny Jordan gave the Rebels the lead. Receiving the ball on the edge of the area, he turned and fired a volley into Strudley's right hand corner.
Slough never looked comfortable at the back, and this was evident as Jamie Jackson called to clear, but Ryan Fenton let the ball go over his head. Jemel Johnson failed to add to his tally, blazing just the wrong side of the angle.
Thatcham took advantage of more poor defending as Slough allowed Dave Asker time to pick his spot from the edge of the area and he sliced the ball into Jackson's top left corner. Thatcham then had a glorious opportunity to go back in front through Flockton's near post header from a corner, but he somehow put it wide.
Just before half time, Slough managed to go 3-2 ahead against the run of play. Pritchard's corner was played in towards Gumbs on the edge of the area, and with the ball not cleared fully, Romeo lined up a strike and fired magnificently into the top corner with Strudley rooted to the spot.
Rebels fans were hoping this goal would set Slough up to go further ahead and kill the game off in the second half, but Thatcham continued creating good chances. Asker broke through the offside trap and found himself clean through but under pressure, he fired straight at Jackson from a central position, and will feel he should have done better.
Slough were still looking to increase their lead, breaking forward wherever possible, and had two glorious chances in the space of a minute. Pritchard flicked the ball over to Platt on the inside right and he fired in a volley which thundered off the underside of the crossbar. Slough followed up and Jordan brought a good save from Strudley with a header low to the keeper's left.
Chances continued to arrive for the hosts, and they took one on 70 minutes, equalizing through Gareth Thomas. A free kick from the right hand side was played to the back post, where it was missed by Jackson and headed back across goal for Thomas to prod the ball home into an unguarded net.
Just four minutes later and Thatcham went 4-3 ahead with Jemel Johnson grabbing his second. He received a long diagonal ball from the right back and cut inside Fenton before firing low across goal into the far corner, despite Jackson getting a touch to the ball.
Slough were desperately seeking a way back into the game, and introduced Glen Harris to replace the more defensively-minded Chris Fermie to give the midfield better attacking options, with Byron Bubb having replaced two-goal Romeo.
With the clock ticking down, the game seemed lost. But then Slough were handed a stroke of fortune with two minutes to go. A Thatcham defender tried to head back to his keeper but got little contact on it, and James Pritchard stole in and rounded Strudley before slotting in from a narrow angle to make it 4-4, sparking jubilation behind the goal.
However there was still another twist in this game as Ashley Vine popped up with the injury-time winner. His header bounced down off the bar and over the line after a free-kick on the right was headed back across goal towards the unmarked dangerman, to deny Slough a replay which they must have felt was on the cards minutes earlier.
Thatcham Town Lineup
1. Paul Strudley, 2. Jamie Leacock (c), 3. Paul Taplin, 4. Gareth Thomas, 5. Sam Rea, 6. Sam Rea, 7. Dean Flockton, 8. Dave Asker, 9. Ashley Vine, 10. Jemel Johnson, 11. Mario Nurse. Subs: 12. Stefan Brown, 14. Ryan Williams, 15. Barry Elmore, 16. Adam Pilling, 17. Steve Hale.