But the Bulls, despite Kris Taylor's late goal, are now on the brink of a return to League Two which they only left last season.
Brighton enjoyed much the better of the first half and could have been well ahead at the interval.
After a couple of Dean Cox shots in the opening exchanges, The Seagulls took the lead after just nine minutes.
Owusu, who was a constant threat to the Hereford defence, headed the ball across goal and Fraser produced a scissor kick as he fired the ball past home keeper Peter Gulacsi.
It was Brighton's first goal in 379 minutes but Fraser had waited even longer - since December 2006 - to find the back of the net.
The lead was almost doubled 12 minutes later when Cox bent a free-kick around Hereford's defensive wall and forced a fine diving save from Gulacsi.
The Bulls defence continued to be forced to repel pressure and it took some good defensive work by Richard Rose to block efforts from Tommy Elphick and Owusu in the lead up to the interval.
Ben Smith and substitute Stephen O'Leary shot just wide for the Bulls early in the second period.
But Brighton soaked up such Hereford pressure as there was without too much inconvenience before extending their advantage 15 minutes from the end.
Substitute Gary Hart neatly found Owusu and the big forward placed his shot beyond Gulacsi.
Hereford found a little late energy and a foul on Febian Brandy brought them a stoppage-time penalty.
Goalkeeper Michels Kuipers saved Steve Guinan's spot-kick but the ball remained in the danger-area before Taylor lashed home a consolation effort from 25 yards.