In doing so, the Rebels inflicted a damaging defeat on sixth-placed Chippenham, who finished with ten men.
The odds were stacked against Slough, with centre-backs Adam New, Guy Hollis and Dan Hicks all unavailable, and Scott Harris, who was set to deputise, also ill. Captain Sean Fraser stepped in to partner Tom Williams and had a fine game, whilst Warren Harris slotted in at right back as Slough reshuffled.
Both teams battled hard to try and play football on the uneven Holloways Park playing surface, and there were chances at both ends. Three minutes in, Matt Smith looked to have got a little flick on a cross which almost deceived Mark Scott, but luckily it was straight at the Slough stopper.
Scott dealt with a free header from a corner easily, and Slough grew in confidence, with a super cross-field pass from Lewis Putman finding Gurkan Gokmen, who shifted the ball inside onto his left foot before bending it just wide of the far post.
Chippenham thought they had taken the lead on 23 minutes, but Smith was adjudged to have been offside as he finished after Andy Sandell's shot had been blocked.
Slough took the lead on 33 minutes when Lee Barney lobbed the ball over goalkeeper Kent Kauppinen, who had advanced out of his goal, from outside the box, as he ran onto a ball played over the top. It was a cool finish from the striker, who notched his sixth of the season.
Chippenham failed to test Scott again in the first half but stepped up the pressure in the second half. The industrious Gokmen crossed in for Mpi, who failed to control the ball and it ran through to Kauppinen. Chippenham's Mark Preece volleyed over before Leigh Henry's shot was easily held.
Warren Harris hooked an effort goalbound which was held by Kauppinen on his goal line, and a minute later Barney narrowly missed his header from Charlie Mpi's cross.
Scott pulled off a crucial save as he got down low to keep out Harry Grant's angled drive, and ten minutes later Alex Ferguson was also thwarted as he tried his luck from outside the box after Ollie Taylor won his battle with Jhai Dhillon by the touchline and passed the ball across the edge of the box.
Chippenham upped the pressure as their search for a goal became more desperate, but Slough's makeshift back four did a sterling job of keeping the visitors out.
Scott got down well at his near post to push Taylor's shot wide, and seven minutes remained when Grant's shot was blocked in the Slough penalty area, leading to appeals for a penalty which fell on deaf ears.
Slough counter-attacked and had chances to kill the game off. Ryan Hope created space for a shot which Kauppinen saved with a leg.
The second half had become a somewhat bad-tempered affair, with the referee stopping play on numerous occasions to penalise players on both sides, and Chippenham substitute Chris Allen was dismissed with a minute remaining after picking up two bookings.
With Chippenham committing numbers to attack, there was still time for Gokmen to have a shot saved after a strong run from Mpi, and Barney worked space in the box but Kauppinen held his effort.
Slough held on to their slender lead and claimed all three points as reward for a gutsy and determined performance.
Chippenham Town Line up
Kent Kauppinen, Alec Fiddes, Leigh Henry, Michael Pook, Greg Tindle (c), Mark Preece, Alex Ferguson, Matt Smith, Alan Griffin, Andy Sandell, Matt Jones. Subs: Luke Ballinger, Harry Grant, Ollie Taylor, Ben Whitehead, Chris Allen.