In a tie delayed to give the Bulls a little more time to recover after their Emirates FA Cup exit to SkyBet League One outfit Fleetwood Town last Thursday night, a crowd of 1,518 braved the freezing cold.
And they saw former Bath City player Billy Murphy rifle home the opener from outside the area in the 18th minute after the ball fell to him following a corner.
Lance Smith made it 2-0 after 27 minutes, the striker doing well to get a shot in and beat Mark Cousins in the Dagenham goal.
However, Jack Deaman diverted the ball into his own goal to give Dagenham hope just 2 minutes later.
The Daggers equalised with 20 minutes remaining when Daniel Sparkes put in a teasing cross on the left and as the ball came back out, Michael Cheek was on hand to slot the ball home.
But Hereford were awarded a penalty 13 minutes from time when Pablo Haysham went down in the box and Mike Symons duly slotted the spot-kick home past Cousins.
Home `keeper Alex Harris made a good save to deny Cory Whitely in the 81st minute but, despite 5 minutes of stoppage time, the Bulls held out to record yet another cup upset.