United were ahead after just 70 seconds when Crewe goalkeeper Rhys Taylor flapped at Kevin Ellison’s corner-kick and the ball hit Dugdale and deflected into the back of the net.
The Alex keeper did better with Ryan Taylor’s turn and snapshot before Shaun Miller fluffed a volley at the other end. Ashley Westwood should have levelled when one-on-one with Jamie Annerson but the Alex midfielder planted the ball wide of the far post.
Annerson then denied Clayton Donaldson with a superb reflex save and former Miller Dave Artell nodded wide before Ryan Taylor headed against the Alex crossbar.
Rhys Taylor dived low to keep out Ellison’s drive as the Millers cut Crewe open while Dugdale flashed a fierce shot just over the crossbar.
Adam Le Fondre had the chance to extend the lead from the penalty spot when Patrick Ada was penalised for handling the striker’s goalbound header.
But Rhys Taylor produced a fine fingertip save to make up for his earlier error and keep Crewe in the game.