Bradford (16) 22
Tries: Sheriffe, Halley, Nero, Menzies
Goals: Sykes 3
Warrington (18) 26
Tries: Briers, Atkins, Anderson, Hicks 2
Goals: Bridge 3
Two-try scorer Chris Hicks is crudely held by Bulls hooker Heath L'Estrange
Challenge Cup holders Warrington were given a real scare at Odsal before two second-half Chris Hicks tries secured a breathless passage to the semi-finals.
After going 12-0 up inside 14 minutes, it briefly looked like Warrington might repeat their stunning 60-4 cakewalk at Huddersfield in the last round.
Bradford rallied with first-half tries from Rikki Sheriffe, Dave Halley and Chris Nero to go in 16-18 at the break.
But Hicks struck twice, Steve Menzies' response for Bulls proving not enough.
Experienced playmakers Lee Briers and Michael Monaghan, architects of Warrington's Wembley triumph against Huddersfield last August, again laid the victory platform.
But the finishing touches were applied by 33-year-old Aussie winger Hicks, in his farewell season in Super League, as Warrington held off a superb effort from a below-strength Bradford to join Leeds, St Helens and Catalans Dragons in Wednesday's semi-final draw.
The Bulls had key scrum-half Matt Orford back after a three-match injury absence.
But they were without Brett Kearney, Stuart Reardon, Jason Crookes and Jamie Langley, forcing coach Steve McNamara to give a debut to 18-year-old Joe Wardle.
And, having revealed that several players defied doctor's orders to play, McNamara's biggest blow was losing another teenager in the closing stages, 18-year-old back row forward James Donaldson, who appears to have torn a cruciate knee ligament, which would signal the end of his season.
Warrington enjoyed a dream start when Orford's kick was charge down and David Solomona slipped in Briers after only two minutes.
Warrington winger Chris Hicks saunters in for the first of his two tries at Odsal
They then doubled their lead when Briers turned from try taker to maker when he timed his pass for centre Ryan Atkins to glide through a gap in the Bulls defence, Chris Bridge again adding the extras, with the second of his three kicks.
But the Bradford fightback began when, on the say-so of the video referee, Sheriffe's try in the corner from Dave Halley's pass was allowed.
Full-back Halley himself then supported a 40-yard break down the middle from hooker Heath L'Estrange to touch down.
The Wolves hit back instantly when Monaghan, man of the match in last year's final, broke clean through the first line of defence and, although his pass was visibly forward, the supporting Vinnie Anderson was allowed to go in under the posts.
But the game was back in the balance by half-time when Bradford centre Chris Nero scored straight from a scrum after Warrington conceded possession from Paul Sykes' re-start.
A tenuous 18-16 interval lead was extended by Hicks, who took Richie Mathers' brilliant pass to cross unopposed at the corner for his first try.
And, after an absorbing but try-less 20 minutes, Hicks looked to have sealed victory when he won the race to get to Monaghan's grubber kick, touching down with barely an inch to spare.
The action continued right to the final whistle, veteran centre Menzies rolling back the years with a solo try 16 minutes from time.
Paul Sykes' third successful conversion pulled the Bulls to within four points of the Wolves, to set up a frenzied finish. But Tony Smith's side hung on by the skin of their teeth.
Warrington coach Tony Smith told BBC Sport:
"We made hard work of it. And there's some areas we need to shape up on if we're to progress in this competition again.
"I knew it was going to be close, as I thought they'd be very desperate and they were.
"We played some really good stuff at times but Bradford wouldn't lie down. They kept on coming and finding ways of fighting back.
"It was fantastic entertainment but very tense. A bit like yesterday at Headingley but this time with tries."
Bradford Bulls coach Steve McNamara said:
"The players are pretty distraught. There are some tears in there, they invested so much in that game.
"In many ways it was a classic cup-tie. We couldn't at any stage get our noses in front and, if we had have done, the result might have been different.
"The team have put a lot into this week and into the game. There were some people out there who had no right to be on the field."
Bradford Bulls: Halley, Sheriffe, Menzies, Nero, Platt, Sykes, Orford, Scruton, L'Estrange, Lynch, Whitehead, Hall, Donaldson.
Replacements: Wardle, Worrincy, Godwin, Kopczak.
Warrington Wolves: Mathers, Hicks, King, Atkins, Riley, Briers, Bridge, Morley, Monaghan, Harrison, Solomona, Westwood, Grix.
Replacements: Higham, Cooper, Mitchell, V Anderson.
Referee: Richard Silverwood (Dewsbury).