Rebels were two goals down heading into the interval, but a headed goal from Adam Martin just before the break gave his side hope.
Kyle Anthony was dismissed for two bookable offences in the second half, and despite creating a number of good chances, Slough could not find an equaliser and slipped to another defeat.
Biggleswade, beaten at Holloways Park recently, had an early chance with a shot deflected just wide of the post, whilst Adam Martin hit a snap-shot from the edge of the box which Michael McEntegart did well to turn it wide for a corner.
The game opened up into an end-to-end affair, and Slough, playing better football than in recent games, went close again when a long Ashley Deeney throw was headed on by Martin, with Burnell just unable to make contact sliding in at the near post.
The hosts took the lead with 14 minutes played, when a superb cross-field ball from Lee Allinson found Brett Donnelly in acres of space, and he made no mistake.
Slough tried to fight back quickly and Ryan O'Toole fired a shot over, but the Waders doubled their lead in the 32nd minute. A long throw from Max York found Steve Gentle’s head, and he flicked on to Allinson, who struck the ball sweetly first time from the edge of the box, beating Warrington in the goal.
The Rebels had been playing well, but found themselves two down. Steve Bateman's side rallied, and after Burnell had a close-range effort well saved, Stuart Swift and Jake Parsons were also thwarted as they had superb efforts blocked.
But on the stroke of half time, Slough gave themselves hope when Adam Martin headed home unchallenged from a Swift corner.
Slough had a let-off early in the second half as Gentle missed his kick in front of goal, whilst the Rebels squandered a good chance to equalise when Burnell shot inches wide of the post.
Slough kept the pressure on, but the Waders continued to repel them and carried a threat on the counter attack. Parsons forced another save from McEntegart before Slough were dealt a blow when Kyle Anthony was shown a second yellow card after pulling back a striker.
With the game stretched, McEntegart made a superb save from O'Toole and Warrington also made a good low save, but time ran out for Slough as they slipped to a fourth defeat in just nine days, their promotion hopes beginning to look a distant memory.
Biggleswade Town Line up
Michael McEntegart, Max York, Peter Gatti, Kevin Kilroy, Steven Gentle (c), Lee Allinson, Craig Daniel, Joe Reynolds, Brett Donnelly, Luke Knight, Jon Woolf. Subs: Paul Barnes, Ricky Case, Matt Clifford, Alex O'Brien, Leon Osei.