A minute's silence in tribute to Diana, the tragic Princess of Wales, was followed by uproar when Randall's goal three minutes later from 30 yards' range flew over the Diamonds' wall and into the top left hand corner without anyone even moving.
It was, however, a lead that looked increasingly frail, especially in the second half when Diamonds' superior teamwork, speed and workrate gave them complete control.
The battle of the managerial Brians - ex-West Brom manager Talbot and his old Arsenal team-mate McDermott, still learning the job - underlined the edge provided by experience.
Talbot's highly motivated and imaginative side shrugged off two open-goal misses by Jamie Ndah and the net was way off the radar of Kenny Cramman, Michael Mison and Carl Alford when they were presented with chances to level the game before the break.
Only the ubiquitous Gary Brazil, who seems to be involved in everything positive the Rebels do going forward, could match the creative skills of the Diamonds forwards.
But lack of real support scuppered any promising Slough moves while Diamonds always hunted in packs.
The midfield partnership between Slough's big signings Danny Bailey and Corey Browne has yet to materialise. Instead it was ex-Peterborough midfielder Gary Butterworth who had the destiny of the points under control.
Butterworth spread the ball around effortlessly and made the equaliser which came in the 52nd minute.
The Slough defence were unable to despatch a long throw into the box and Butterworth chipped a half clearance back to Darren Collins who stabbed the ball home at the second attempt after Ben Miles had got a hand to his first effort from six yards.
Slough's tendency to carp at the decisions of the officials also contrasted with Diamonds get-on-with-it approach and they almost took the lead from a quickly taken free kick in the 62nd minute from which Cramman struck the crossbar.
Slough went behind in the 75th minute following sustained pressure, Mison threading a low shot from 20 yards inside the far post past the unsighted Miles.
Brazil continued to provide Slough's best chances - a cross which Owusu missed on the line, another palmed over the bar and a volley steered round the post by Cherry.
And Slough's other Trojan, Cliff Hercules, hit the upright from a corner. The Rebels can have no complaints about this result which sees no improvement in a turgid home record which now shows just one win in the last 14 league games.
Rushden & Diamonds Line up
Cherry, Hamsher, Cramman, Mison, Rodwell, Fuff, Crosby, Butterworth, Ndah, Collins, Hackett, Subs: Underwood (45, for Crosby), Alford (31, for Ndah), Green.