Leading marksman Glenn Murray carved out the Seagulls' best opportunities but had efforts turned round the post by goalkeeper Darren Randolph in either half.
Hereford's workmanlike display stopped a run of five successive away defeats but Brighton had a strong penalty appeal waved away late on when Joe Anyinsah tumbled over.
Randolph, on loan from Charlton, came to his side's rescue after 27 minutes by turning a goalbound shot from Murray round the post after a mistake by defender Richard Rose.
Hereford played with increased confidence after the break and midfielder Kris Taylor put an overhead kick over for the struggling visitors.
Brighton rarely threatened although former Rochdale forward Murray forced Randolph to turn his angled shot round the post midway through the second half.