The Bluebirds looked on course for a first home league win since October 10th after scoring twice in seven second half minutes, but the Rebels responded strongly with goals from substitute Ben Harris and then Lee Togwell a minute into time added on.
The first opportunities fell for Chippenham, with Ross Stearn’s shot blocked and Brad Ash's effort going wide.
Better was to come, with a 30-yard piledriver from Joe Hanks being palmed behind by Jack Turner.
Just a few moments later, Josh Jackman wasted the visitors' best chance of the half as he shot across the face of goal but wide of the far post.
Matt Lench's cross found Paul Hodges, who could only volley straight at veteran goalkeeper Jason Matthews, the 45-year-old deputising in goal for the suspended Will Puddy.
Hodges for Slough, after a fine passage of play, and Ash for Chippenham both saw shots blocked, whilst in first half stoppage time Ryan Bird – utilised as a centre-forward – saw a shot was deflected behind.
The second half was only seconds old when Lench burst forward and was unlucky to see his fierce drive rebound back off the inside of the post.
Just three minutes later, Lench drew a fine save from Matthews, the goalkeeper tipping his shot over the bar.
Lench was in the thick of the action again on 53 minutes, going on a storming run into the Chippenham penalty area, only to stumble and see his tame effort blocked.
Slough continued to look the more likely with Hodges shooting wide on the turn on 62 minutes – but just a minute later, Chippenham punished their opponents by taking the lead.
Ed Jones combined with Stearn before crossing low into the box for the experienced substitute Chris Zebroski to fire past Turner.
Matthews then made a superb save from Louis Lomas’ long-range effort, but again Chippenham made Slough pay for some sloppy play with a second goal.
The hosts doubled their advantage on 70 minutes as Joe Hanks capitalised on a loose pass before running clear and blasting the ball past Turner.
A series of corners for the Rebels eventually led to them pulling a goal back. Warren Harris had already headed over before Ben Harris, fresh from the bench, powered in a header from Lench’s delivery.
More chances came for Slough to pull level, with Matthews again rolling back the years to deny the Rebels from another corner at full stretch. Dan Roberts – also called from the bench – also found himself denied by Matthews, with the striker only able to fire wide on the rebound.
But Slough, even after suffering another injury blow to Ben Harris – who himself had to be replaced by Jack McKnight – were not to be denied as Lee Togwell got the faintest of touches to a long Guy Hollis throw to divert the ball into the back of the net.
Chippenham Town Line up
Jason Matthews, Ed Jones, Danny Greenslade, Tom Harrison, Kieran Parselle, Luke Russe, Adam Mann (Alex Bray 75), Joe Hanks, Ross Stearn, Marlon Jackson (Chris Zebroski HT), Brad Ash (Callum Gunner 66). Unused substitutes: Ethan Hill, Spencer Hamilton.