Travel-sick Bradford slipped to their third away defeat in a week as Crewe prevailed at Gresty Road.
Bantams boss Phil Parkinson made three changes to the team that started Saturday’s defeat at Aldershot Town, though all were enforced.
In came Lee Bullock after shaking off a virus and Deane Smalley across the middle, while Matt Fry replaced shoulder injury victim Marcel Seip at left-back.
Craig Fagan and Ritchie Jones both had to miss out as they had been taken ill.
The opening quarter of the game belonged to the hosts. Luke Murphy fired a shot inches over the top, before the Alex went in front in the 21st minute.
Ashley Westwood’s dangerous cross into the box struck the arm of Andrew Davies and referee Mr Whitestone immediately pointed to the spot.
Harry Davis made no mistake as he slammed his penalty-kick past Jon McLaughlin.
The Bantams began the second half brightly and James Hanson volleyed over, before loan signing Chris Dagnall thought he had equalised.
He bundled the ball home in the 65th minute, but an earlier foul by Hanson meant the effort was ruled out.
The 400-strong travelling army tried to rally the Bantams, and with 20 minutes remaming a positive double substitution from Parkinson brought on David Syers and Nahki Wells.
Wells and the unlucky Dagnall both tested goalkeeper Steve Phillips as the game edged towards stoppage-time, but Crewe managed to hold on for a precious victory.
Crewe Alexandra: Phillips, Tootle (Brown 44), Davis, Westwood, Artell, Dugdale, Murphy, Leith-Smith, Moore, Shelley, Powell. Unused substitutes: Martin, Bell, Miller, Clayton.
Bradford City: McLaughlin, Ramsden, Fry, Oliver, Hanson, Smalley (Wells 71), Ravenhill, Dagnall, Bullock (Syers 71), Reid, Davies. Unused substitutes: Annerson, Kozluk, Atkinson.
Referee: D Whitestone (Northants).