The Daggers were buzzing with confidence after thrashing Chester and they dominated the opening period but the visitors had the first chance.
Striker Garry Thompson fired straight at keeper Tony Roberts after losing his marker on the edge of the box.
Full-back Danny Adams came close for the visitors in the eighth minute when his fierce 25-yard strike was deflected over the bar off a Daggers defender.
The home side weathered the storm and nearly caught Morecambe with a sucker-punch in the tenth minute when Glen Southam saw his clever lob drop narrowly over the bar.
Shrimps keeper Shwan Jalal handed the home side a gift in the 20th minute when he let a weak shot from Peter Gain go through his legs to give the Daggers the lead.
Ben Strevens should have made it 2-0 three minutes later when he thundered a shot wide from eight yards after latching on to a loose ball in the Morecambe box.
Daggers striker Jon Nurse came closer still in the 31st minute when he volleyed narrowly wide from 15 yards after a clever lay-off from Strevens.
The Daggers upped the pressure after half-time with Nurse forcing a fine save from Jalal with a powerful shot from the edge of the box.
Shrimps frontman, Matthew Blinkhorn, should have equalised in the 60th minute when he blasted high and wide from ten yards after shrugging off Daggers skipper Anwar Uddin.
Gain almost caught Jalal napping again with a speculative 35-yard strike but the keeper managed to scramble the ball to safety.
Blinkhorn wasted another opportunity in the 81st minute when he volleyed high over the bar after ghosting into the Daggers penalty area.
Strevens made sure for the Daggers in stoppage time when he ran on to a pass from Roberts and rounded Jalal before firing into the net.