Slough were under pressure from the off, and Mark Scott made a good save to keep the scoresheet blank on four minutes.
Garyn Preen, Hungerford's most dangerous attacking player on the day, fired over under pressure minutes later. Slough gradually settled and Ryan Hope was unlucky not to get the ball under control for a shot at goal as he received possession on the edge of the box.
Hungerford appealed for a penalty as Stefan Brown went down under the challenge of Guy Hollis, but the referee waved play on.
The Rebels struggled to trouble a resolute Hungerford defence, with set piece delivery generally disappointing in the first half. The returning Charlie Mpi was well tackled as he found himself in a race with defender James Rusby from a ball played over the top, and Nathan Webb shot well wide as he picked up on the loose ball.
Mpi sidefooted an accurate inswinging free kick from Hope over the bar six yards out, which was to prove costly as Hungerford took the lead three minutes later. Preen capitalised on a mistake from Jhai Dhillon, who let the ball run across him, and the forward beat Scott at his near post, the ball flashing past the keeper and appearing to catch him by surprise.
Slough found themselves a goal down heading into the break after what was a fairly even first 45 minutes, and their cause was not helped with Hollis unable to continue as he felt the effects of a chest infection. He was replaced by 17 year old Adam New.
Mpi stretched to head another Hope free kick well over, and shortly after Slough had a let-off when Preen's mishit shot came back off the inside of the post before being gathered by Scott.
Scott was tested as he tipped over Gary Horgan's header from a corner, whilst the Rebels were yet to work Paul Strudley in the Hungerford goal with over an hour played.
Preen blasted across goal and wide after a free kick was played into the defensive wall, and Slough finally had their first effort on target on 68 minutes, but Dhillon's shot from outside the box was an easy save.
Almost immediately, Hungerford moved play up the other end of the pitch and doubled their lead. Preen got clear and squared the ball across for substitute Mark Draycott, who had the simple task of sidefooting home from a matter of yards out.
The visitors looked comfortable with their two goal cushion, and continued to keep a shot-shy Slough at bay. Gurkan Gokmen managed to squeeze in a shot which bobbled harmlessly through to Strudley, but the clearest chance for the home side came in the second minute of time added on when New's header was blocked on the line and Warren Harris' shot on the rebound was saved at point blank range by Strudley, who spread himself to preserve his clean sheet.
Hungerford Town Line up
Paul Strudley, Gary Horgan, Rhys Tyler, James Rusby, Matt Day (c), Scott Rees, James Clark, Ian Herring, Stefan Brown, Garyn Preen, Alan O'Brien. Subs: Harry Goodger, Mark Draycott, Mike Jones, Diak John, Jon Boardman.