Dave Tilbury took advantage of a Ricky Perks fumble to give the visitors an early lead, but Sean Sonner equalised after a howler from Burnham keeper Chay Morris left him with an open goal.
In the second half, Chris Seeby headed the Rebels ahead, but Ryan Bird equalised in controversial circumstances before Dan Willment headed the winner with 15 minutes remaining after more poor defending.
Eager to put the last few results behind them, Slough handed former Cambridge United striker Ben Abbey a full debut, and the Rebels looked to attack from the off. Steve Sinclair hit the side netting from 15 yards early on at a bitterly cold Holloways Park.
But before both sides had even settled into the game, a horrendous mistake from Ricky Perks handed the opening goal to basement side Burnham in the ninth minute. A simple-looking catch was spilt and the ball bounced towards the goal-line, and Dave Tilbury was credited with turning the ball over the line.
Burnham clearly looked fired up for the contest and the opening goal only served to make the visitors hassle the Rebels even harder, pressing all over the pitch and not allowing the home side any time on the ball, leaving the long ball the only option on many occasions.
The Blues went close again before Paul Coyne volleyed over Slough’s first real shot from a quickly-taken corner.
On the half-hour mark, Slough equalised in similarly fortuitous circumstances to the visitors. Sean Sonner reacted quickly to latch on to a ball played over the top, and with Chay Morris horribly exposed and committed to trying to meet the ball, Sonner easily got to the ball first, rounded the keeper and finished with ease.
With Slough now in the ascendancy, Swift had a long shot easily saved by Morris – but the Rebels could not find a vital goal before the break.
It looked as if this may not matter so much as Slough emerged the brighter side in the second half. Steve Sinclair looked the most likely to create a goal, and the little winger played a quick ball across the face of goal, but just too far ahead of Ben Abbey.
Ten minutes into the half, Slough took the lead for the first time as Chris Seeby powered in a header from Stuart Swift's inswinging corner, and the ball squeezed past Chay Morris and just over the goal line.
But yet again, Slough threw away a winning position. On 71 minutes, Yashwa Romeo hit the post from a tight angle with Ricky Perks appealing for an apparently blatant handball by the former Slough striker. Time seemed to stand still as Ryan Bird followed up to finish into the empty net.
And with 15 minutes left, Perks was beaten again by a looping header from Dan Willment with the goalkeeper badly out of position, as Slough’s defensive frailties were exposed again.
Jamie Jarvis had Chay Morris to thank for keeping his side ahead, as the keeper redeemed himself for his earlier aberration by superbly pushing Stuart Swift's free kick away, when it looked destined for the top corner.
Sam Bateman blazed a clear chance well over minutes later as Burnham maintained their excellent record against Slough in recent seasons, and ensured the Rebels’ desperate recent run of results continued.
Burnham Line up
Morris, Tugwell, Hudell, Tilbury (c), Allaway, Willment, Lumsden, Chandiram, Romeo, Bird, Hicks. Subs not used: Ruchell, Kemp, Lofthouse-Smith, Crane, McNeil.