Alex substitute Byron Moore rifled home after the break to cancel out Nurse's first-half stunner, but the Daggers struck back instantly through Benson to take home three well-deserved points.
Alex hearts were in mouths on the quarter-hour mark when Solomon Taiwo struck a 25-yard free-kick that whistled millimetres wide.
Sixty seconds later Daggers keeper Tony Roberts tipped Calvin Zola's header onto his crossbar before Antony Elding awkwardly headed over when he met Billy Jones's free-kick at the far post.
A mix-up between Crewe keeper David Button and defender Danny O'Donnell left Benson with an open net but the angle became too acute to find the target.
However, the Daggers took the lead two minutes before the break when Nurse beat Button with a perfect 35-yard half-volley. Moore stepped inside Mark Arber and drilled the ball high into the far corner of the net to level after 71 minutes.
But Benson barged his way past two defenders before rounding Button and rifling home from close range just two minutes later.