The Alex striker clinically punished poor defending by the Bantams, as did Bradford-born Clayton Donaldson and Danny Shelley who netted the other goals to give the Railwaymen a commanding lead.
Crewe had taken the lead on 12 minutes when a poor attempted pass by David Syers to Luke Oliver allowed Shaun Miller to nip in and steal possession.
Then, as Jon McLaughlin rushed from his line, the Crewe striker deftly flicked the ball past the City goalkeeper before rolling the ball into an empty net.
Bradford responded midway through the half with a couple of decent chances as Speight brought a sprawling save from Rhys Taylor and then James Hanson fired against a post.
The equaliser came midway through the half when Speight was tripped by David Artell and referee Graham Salisbury pointed to the spot.
Speight then got to his feet to send Taylor the wrong way with the resulting penalty.
Within 11 minutes, however, the Bantams were behind again after a sweeping move that saw Byron Moore race down the right and centre for Miller to convert.
Sixty seconds later, it was 3-1 when Donaldson showed a neat touch to turn on the edge of the area before drilling a shot beyond McLaughlin.
City’s misery was compounded when McLaughlin allowed a shot from Shelley through his hands on the stroke of half-time.
Leading 4-1 at the break, the Railwaymen cruised through the second half but still managed to add one more goal.
That came on 67 minutes when Matt Tootle got to the by-line before sending over a deep cross that the home side could not clear.
The ball eventually broke to Shelley, whose shot deflected into the net to ensure Bradford suffered their worst defeat since Sheffield United triumphed here in Division One almost nine years ago.
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