Jon Parkin should have put the home side ahead after just two minutes after a poor back-pass fell at his feet but he placed his shot too close to Steve Phillips.
The League One club enjoyed plenty of first half possession without converting it into chances and were made to pay six minutes from the break.
It was a neat counter-attack from Fleetwood, with Liam Hogan laying the ball on for David Ball to produce a sublime finish from 20 yards.
Fleetwood doubled their advantage before half-time. Gareth Evans was released down the left and crossed for Parkin, with his header being deflected in.
Victory was assured a minute after the restart as centre-half Nathan Pond, turning 25 yards out, released a low shot which skimmed past Phillips.
Crewe tried desperately to battle back, Kelvin Mellor's shot being tipped wide and Chucks Aneke having a weak effort from the edge of the area.
But nothing seemed to be working for the champions and Fleetwood added a late fourth as sub Matthew Blair slid in to convert from a Stuart Murdoch cross to book their place in the area quarter-finals for the first time.