The match went with the formbook as the Daggers - winless in five before kick-off - were undone by two goals in as many minutes early in the piece.
The Shrimps took the game away from their hosts in the blink of an eye.
Dagenham barely had time to collect themselves after the shock of Diarmuid O'Carroll's opener before they were trudging back to the halfway line when Stewart Drummond fired in a second two minutes later.
The first came as a result of a set-piece and a collective failure by the home side's defence to pick up their men in the box, allowing O'Carroll to nod the ball in off the inside of the post.
The visitors ventured forward immediately after regaining possession from the restart and while their initial raid was cleared, the ball came out to Drummond on the edge of the box.
The midfielder took the ball calmly on his chest before unleashing a fierce left-footed volley that took a slight deflection on its way into the net.
Drummond could have put his team out of sight minutes later but he blazed over from five yards out.
After a sluggish start, the Daggers began to look more threatening as the game wore on.
The tireless Jamie Guy, making his first start for the club, did well to set up Doug Loft, only for the midfielder to skew his shot horribly wide as he bore down on goal.
Guy, clearly looking to impress, was doing just that as some excellent work created the chance for Matt Ritchie to shoot, but he drew a superb diving stop from Barry Roche.
The hosts surged forward but it was Morecambe who came closest to scoring as Wayne Curtis was denied by Tony Roberts after bundling his way through.
Another rasping drive from Scott Griffiths was kept out by the considerable frame of Roche, as the visitors held out for all three points.