The visitors looked nervy from the start with rookie striker Dominic Green testing keeper Anthony Williams with a powerful strike from the edge of the box in the fifth minute.
Wrexham's shaky backline were found wanting again in the 19th minute when Daggers midfielder Jon Nurse raced past defender Shaun Pejic to power a thunderous shot straight at Williams from 20 yards.
The Dragons defence finally cracked in the 24th minute when centre-back Steve Evans headed a deep cross from Nurse past his own keeper to put Dagenham ahead.
Midfielder Glen Southam came close to doubling Dagenham's lead in the 37th minute when he powered into the Wrexham box before curling in a powerful strike which Williams did well to push over the bar.
Southam missed a sitter on the stroke of half-time when he failed to connect with an inviting cross from Green with the goal at his mercy.
Daggers midfielder Shane Huke tried his luck from long-range in the 52nd minute but his thunderous strike flew high and wide of goal.
The Dragons almost caught Dagenham with a sucker punch in the 72nd minute when Chris Llewellyn latched on to a Sam Aiston through ball only to shoot straight at Daggers keeper Tony Roberts from the edge of the box.
Robbie Garrett came even closer to grabbing an equaliser in the 78th minute when he volleyed inches over the bar after pouncing on a loose ball on the edge of the box.
The home side soaked up the pressure and sealed the win in the 85th minute when substitute Ben Strevens hammered a 30-yard thunderbolt into the top corner of the net.
Daggers rubbed salt into the wound in stoppage time when substitute striker Jamie Taylor fired a low shot past Williams to make it 3-0 with his first touch of the ball.