Paul Benson and Solomon Taiwo grabbed the first-half goals which ensured the Londoners a tie at the Walkers Stadium against Leicester City.
Dagenham were fortunate to earn a replay after escaping Edgar Street with a goalless draw a fortnight ago and quickly took advantage of their second bite at the cherry.
The League Two side bossed the opening stages of the game and duly took a 17th minute lead when Benson collected Sam Saunders' cross and swept his shot low into the bottom left-hand corner.
And their place in the second round looked all but assured 16 minutes later when Taiwo latched onto Benson's flick and rifled a 25-yard shot past Craig Samson.
But John Still's men immediately took their collective eye off the ball and allowed their opponents back into the contest less than 60 seconds later.
The Daggers defence backed off and allowed Kris Taylor a free shot at goal from the edge of the area and he gave Tony Roberts little chance.
Rather than attempt to build on their precarious lead, the home side retreated into their shells as the start of the second half, and survived because of Hereford's wastefulness in possession rather than their excellence in defence.
Steve Guinan did test Roberts with a decent effort with 20 minutes remaining, but the veteran goalkeeper demonstrated expert handling to hold on to the shot from distance.