Anwar Uddin, who spent a loan period at Edgar Street last season, netted the decisive goal after 34 minutes and, despite heavy Hereford pressure in the second period, the visitors held on for their win.
David Brown should have opened the scoring on six minutes when he nipped in to intercept after defender Danny Foster tried to guide a long goalkick back.
Goalkeeper Tony Roberts was alert to the situation, though, and snatched the ball off Brown's toes as the forward failed to control.
Hereford continued to have the better of play but strong Dagenham defensive work prevented them from getting in many serious shots to trouble the veteran keeper.
Ten minutes before the break the visitors took the lead from their first meaningful effort on goal.
Glen Southam swung over a free-kick and Uddin glanced a header into the corner of the net.
Roberts then produced good stops to repel long-range efforts from Jamie Pitman and Rob Purdie as Hereford tried to hit back before the break.
Injuries forced both sides to reshuffle at the break but Hereford pressed strongly and had the lion's share of the possession.
But despite forcing nine corners in the half, they failed to put Roberts under much pressure and the Essex side themselves broke out to create a couple of good openings through Chris Moore.
Hereford's best effort of the half came midway through the five added minutes at the end when on-loan Joe Anyinsah sent a 25-yard effort slashing just wide of an upright.