Substitute Luke Murphy gave the Alex hope with a second-half strike, but the visitors held on.
Crewe rookie Carl Martin was thrown into central defence after just 20 minutes when Danny O'Donnell hobbled off, and the Cherries were in front a minute later. Ashley Westwood cleared off the line from Lee Bradbury's far-post header, but Brett Pitman whipped in a cross from the resulting corner and Fletcher headed in unmarked.
Joel Grant linked up well with Calvin Zola, only for Cherries goalkeeper Shwan Jalal to produce a superb double save from the 16-goal hitman.
Rhoys Wiggins' vital block kept out Grant again, and the vistors extended their lead in the 63rd minute when Alex goalkeeper Steve Phillips could only parry Pitman's shot to the feet of Robinson for an easy finish.
Three minutes later Murphy volleyed home from 20 yards to offer Crewe a lifeline, but Bournemouth held on for three points.