The Bantams scored twice in six second half minutes to condemn Graeme Lee to his first defeat on home soil as Pools boss.
Pools had chances in the first half through striker Marcus Carver before Joe White struck the bar after the break.
But it was the visitors who hit the front when Matt Foulds capped a swift counter-attack before Yann Songoo added a second.
And here is how Pools were rated in defeat.
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1. Ben Killip - 7
Excellent stop to deny Cook in the first half. Might have felt he could do better with the second after getting a hand to it. (Credit: Will Matthews | MI News)
Photo: Will Matthews
2. Jamie Sterry - 5
Looked a little jaded up against Sutton. Out of position for Bradford’s opener. Picture by Martin Swinney
Photo: Martin Swinney
3. Neill Byrne - 6
Did well against Cook including a key defensive header in the first half but part of a number beaten at the near post for Bradford’s second. (Credit: Michael Driver | MI News)
Photo: Michael Driver
4. Timi Odusina - 7
Brilliant in the first half but came under a bit more pressure in the second resulting in a yellow card. Picture by FRANK REID
Photo: Frank Reid