Billy Clarke's first-half goal was enough to see the Bantams past the Railwaymen, who relied on scores elsewhere to keep them in the division.
Crewe, who had been unbeaten in seven games heading into today, started in scrappy fashion as both Clarke and Jon Stead fired over for the visitors.
Clarke then gave the Bantams the lead when a free kick arrived at his feet. The Irish forward produced a neat turn and shot that left Paul Rachubka beaten.
At one stage in the second half, Crewe languished in the bottom four as other side in the relegation zone looked to be climbing out.
In the end, Crewe were saved by Gillingham's late rally, which saw Notts County drop out of the division.