Playing in front of 31,054 - a record for the competition outside the overall final - City had the better of the first half on Tuesday night but could not find a way past an in-form Steve Phillips.
The Crewe keeper thwarted Stephen Elliott on three occasions, while the visitors snatched a 52nd-minute lead.
Bradden Inman led a swift counter attack forward and applied a devastating finish with his left foot after a square ball from Byron Moore.
Phillips was called upon again to prevent a deflected clearance finding the net before Carl Baker smashed the foot of his near post.
Inman then struck again, producing a magnificent solo run before firing low into the net from 18 yards, 12 minutes from time.
Ajay Leitch-Smith added a well-finished third with just six minutes left to make Crewe hot favourites for a trip to Wembley.