The South Coast club have a 100 per cent record in cup competitions so far, including a shock victory against Manchester City in the Carling Cup.
But their league form has deteriorated alarmingly after a promising start and they have now gone eight games without a win.
Hereford, held together at the back by Bruno N'Gotty, set their stall out from the early stages. They were quick to get men behind the ball and content to ask the home team to break them down.
Brighton's first-half efforts petered out after an encouraging opening.
Hereford captain, Karl Broadhurst was a little fortunate in the 21st minute when referee Mike Russell decided his foul on Glenn Murray was just outside rather than inside the penalty area.
Murray, Brighton's leading marksman, was denied a few minutes later. Richard Rose inadvertently headed him into space, but his low, angled drive was diverted for a corner with his foot by keeper Darren Randolph.
Hereford showed a bit more attacking intent after the break as the Brighton supporters grew increasingly restless.
Kris Taylor had a 25-yard free-kick tipped over the bar by Michels Kuipers and Brighton almost gifted the visitors a late goal when Adam Virgo sold Kuipers short.
The keeper could only block the ball into the path of substitute of Nick Chadwick, but his shot could not find its way through a congested box.
Brighton must have sensed it was not going to be their day when referee Russell waived aside appeals for a penalty as Joe Anyinsah appeared to be bundled off the ball in trying to reach a low cross by substitute Kevin Thornton.