The result capped a frustrating six days for the Rebels, whose efforts were not rewarded against Frome and St Neots, and Jon Underwood and Neil Baker's side deserved more than a defeat from this match at Bulpit Lane, where Hungerford ground out a result in a tight game.
The decisive moment came in the tenth minute when James Rusby was left unmarked at a corner, and he headed into the net from Alan O'Brien's delivery.
Scott Harris had just gone close, but his attempted overhead flick was cleared at the back post.
Slough struggled to hold onto the ball after going behind, but managed to fashion chances, with Dan Hicks heading an effort at Paul Strudley, and later the keeper pulled off a good save when he tipped over Lewis Putman’s dangerous cross.
Hungerford went in to the half time break with a deserved lead, but Slough had the better of the second half and left the hosts hanging on at the end.
The home side initially started well, with Stefan Brown flicking a cross from O'Brien just over, but Slough grew in confidence and forced Hungerford back into their own half.
Lewis Putman was often Slough's main threat as he used his pace and trickery out wide on the left, and on numerous occasions he worked space for dangerous crosses which Strudley or his defenders had to deal with.
Ultimately, there was no lack of effort or commitment from the Rebels, but they were unable to break through a well organised Hungerford defence, and the home side were able to soak up the pressure and hold on to their single goal lead.
Hungerford Town Line up
Paul Strudley, Gary Morgan, Rhys Tyler, James Rusby, Matt Day (c), Ian Herring, James Clark, Mike Jones, Stefan Brown, Mark Draycott, Alan O'Brien. Subs: Harry Goodger, Ollie Harrison, Garyn Preen, Jack Stirland, Scott Rees.