The result will be particularly galling for the visitors, who took the lead after 49 minutes when Brett Donnelly headed a cross past James Warrington.
In truth, Slough had not looked like scoring, but did so with their first meaningful attempt on target with five minutes remaining.
Jerome Anderson was handed his first start, but struggled to make an impact alongside Sean Sonner and Steve Sinclair in attack - however he made a telling contribution when, after beating his man on the right, he crossed in to the middle for the onrushing Danny Murphy to superbly side foot past Anthony Ender in the Biggleswade goal to give Slough what looked like an undeserved point.
But Slough's tails were up, and after surviving a scare when Harry Hunt's low ball into the box was cleared, the Rebels won a free kick in a dangerous position two minutes into stoppage time. Lloyd Owusu came on to make his final Slough appearance before departing for Australia, and his impact was immediate, although he was not directly involved in the goal. Anderson worked space for a shot which was blocked, but substitute Ben Abbey reacted quickly and fired a shot through a crowd and past the unsighted Ender to seal the three points and keep his side in touch at the top.
After a hugely disappointing performance and result in midweek, a response was needed, but Slough started sluggishly and Alex O'Brien had the first effort of the game, but Warrington pushed it over comfortably.
Slough did put together a promising move as the ball was worked to Burgess, but his ball across from the right was missed by Anderson, who was in an offside position anyway.
Biggleswade enjoyed the greater share of possession and Slough struggled to make much of an impact - and the visitors had a better chance to take the lead, but a shot was hooked wide from the edge of the box with Warrington beaten.
Alex O'Brien put another shot inches just wide shortly before half time, but their frustration was to be short-lived as Donnelly headed his side in front four minutes into the second period after Murphy had been beaten on the right hand side.
Stuart Swift put a free kick a foot or so wide, but that was as good as it got for Slough for long periods, and Biggleswade should really have extended their lead, Warrington getting down well to deny Craig Daniel's effort.
Steve Bateman made changes, with Dan Burnell and Ben Abbey introduced, but still Biggleswade dealt with everything thrown at them by Slough, until five minutes from time when Anderson broke free of the shackles and managed to tee up Murphy.
Ben Abbey's late heroics kept Slough level on points with St Neots at the top, at least until Tuesday evening, and sparked joy amongst the Slough contingent - along with a huge feeling of relief after Biggleswade had looked in control for large parts of the match.
Slough Town Line Up
- 1 James Warrington
- 2 Simon Sweeney
- 3 Danny Murphy 15
- 4 Adam Logie
- 5 Nathan Bowden-Haase
- 6 David Woozley
- 7 Steve Sinclair
- 8 Ollie Burgess 14
- 9 Jerome Anderson
- 10 Sean Sonner 12
- 11 Stuart Swift
Biggleswade Town Line up
Anthony Ender, Peter Gatti, Darryl Coakley, Liam Folds, Steven Gentle, Craig Rydeheard, Craig Daniel, Alex O'Brien, Brett Donnelly (Harry Hunt), Jonathan Woolf, Ashley Fuller. Subs not used: Alaa Oujdi, James Freshwater, Kevin Kilroy.