The Wolves, who had won seven of their previous eight games before today, had to make do without several key players and also lost Lee Briers with an apparent hamstring injury.
Converted scores from Chris Nero and Glenn Morrison, plus a penalty and drop goal from Paul Deacon, put the Bulls in charge.
And despite a second-half rally from the hosts which saw Simon Grix and Chris Hicks cross, a further try from Terry Newton made sure of the win.
Adrian Morley and Ben Westwood were missing for the Wolves through international duty, as was Bradford's Sam Burgess.
Mickey Higham (torn bicep) and Matty Blythe (ankle) also did not figure for the hosts, while Paul Sykes (hamstring) missed out for Bradford.
The Bulls, who went down 58-22 at home to the Wolves in April, were out of the traps quickly and dominated the early stages, but had a Steve Menzies score correctly chalked off by the referee.
The Bulls knew they were in a battle when loose forward Jamie Langley had to leave the field with a cut to his head and Terry Newton rolled away from a tackle in pain.
Both remained on the pitch but the Wolves, who lost Briers to what looked liked a hamstring injury, grew in confidence and went close when Matt King spilled the ball as he stretched for the line.
The first try came in the 15th minute when Menzies should have been bundled into touch on the right.
The Bulls centre somehow kept the ball alive and after the ball was thrown wide stand-off Ben Jefferies kicked to the left for Nero, who stepped past a couple of Wolves defenders on his way to the line.
Warrington thought they had done enough to hold the centre up but the try was awarded.
The frantic pace gradually slowed as the heat became a factor and Warrington gave away a soft penalty seven minutes before the break for what the referee deemed a dangerous tackle.
Deacon slotted home from 30 metres and then dropped a goal as the half-time siren sounded to give the Bulls a 9-0 lead.
Bradford set the tone for the second period when replacement Elliott Whitehead crossed after a neat pass from Deacon but his effort was ruled out.
Paul Rauhihi broke well for Warrington but he looked to force the pass and Bradford pounced on the loose ball.
The Bulls swept down field and after working the ball from left to right, Morrison sliced through the Wolves defence and stepped past Richard Mathers before touching down.
Deacon was on target with the conversion but the home side pulled a try back in the 54th minute.
Chris Bridge managed to find Grix in space out wide and he stepped inside and raced under the posts, avoiding some poor Bradford tackles in the process.
Bridge made no mistake in adding the extras.
Newton scored a third Bradford try in the 65th minute when Dave Halley offloaded well. The Bradford hooker charged for the line and crashed over for an easy score.
Hicks grabbed his 16th try of the season when he forced his way over in the corner at the death but it was too little, too late for Warrington.
Warrington (0) 10 Tries: Grix, Hicks. Goals: Bridge
Bradford (9) 21 Tries: Nero, Morrison, Newton. Goals: Deacon 4. Drop Goals: Deacon
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