Warrington had to come from behind at the Halliwell Jones as they beat Bradford to book their place in the Challenge Cup quarter-finals.
Adrian Purtell's early try hinted at a shock, but Joel Monaghan helped turn it round with three of the six tries for the 2009 and 2010 Wembley winners.
Brett Hodgson, who kicked three goals, Chris Riley and Ryan Atkins grabbed Wire's other three tries.
Keith Lulia and Olivier Elia replied, but Monaghan then completed his treble.
The Wolves were not at their best but ran in four tries in a 14-minute spell either side of half-time to establish control of a scrappy fifth-round tie.
And home coach Tony Smith pinpointed the impact of substitute forwards Ben Harrison, Simon Grix, Garreth Carvell and Mike Cooper as the deciding factor.
After being hit by Purtell's converted 12th-minute try, Warrington were back level on 24 minutes when Lee Briers' long pass allowed Monaghan to gallop over in the right corner.
But, after sterling defence from the visitors in keeping out Wire, it was two very soft tries that earned the home side the lead just before the break.
Wrongly sensing an infringement following a stray pass, the Bradford defence simply disappeared as Hodgson casually picked up the loose ball and jinked his way through.
And there was another gift right on the whistle when Bryn Hargreaves spilled an attempted pass at Riley's feet and the winger went half the length of the field to score under the posts.
Two tries early in the second half by Monaghan and Ryan Atkins stretched Warrington clear, only for Keith Lulia to score Bradford's second try and pull the deficit back to 26-10.
Wire were then reduced to 12 men after Chris Hill's sinbinning for persistent infringement, the last in a succession of penalties conceded by the home side.
And Elima's try gave Bradford late hope, but Monaghan went in at the corner, Hodgson adding the touchline conversion, to seal victory.
Bradford coach Mick Potter was especially delighted with the performance of half-back Luke Gale, who had a major influence on his return from a two-month absence with an ankle injury, also kicking two of his side's three conversions.
This year's Challenge Cup quarter-final draw will take place on Tuesday, 1 May at 0820 BST, BBC Radio Five live.
FROM THE DRESSING ROOM
Warrington coach Tony Smith:
"In the Challenge Cup sometimes you can win ugly and get through. But I thought our bench was terrific. They gave us some real good energy.
"It was the best I've seen Benny play for some time. He really lifted us and Grix was outstanding when he came on as second hooker.
"Bradford are a good team. They play field position very well, they kick and chase very well and they're a good defensive team. You have to work hard and have some patience against them.
"It was very frustrating for both teams. It was the slowest ruck area since the last time we had the same referee. It was messy, slow and frustrating to watch."
Bradford coach Mick Potter:
"We were in that game. They got the better of us just before half-time and we had a mountain to climb in the second half. When you make errors against Warrington they can burn you.
"There wasn't any lacking in effort. They gave everything, I couldn't ask for any more. There are some good signs there for us as a team.
"It was a big ask for Luke Gale but he acquitted himself very well. I thought he was very good.
"He added value to the side. His kicking and his running game was good and he will get better. It was a tough one for him to come back in."
Warrington: Hodgson; Riley, Bridge, Atkins, J Monaghan; Briers, O'Brien; Hill, Higham, Wood, Waterhouse, Westwood, McCarthy.
Replacements: Harrison, Grix, Carvell, Cooper.
Sin Bin: Hill.
Bradford: Pryce; Purtell, Whitehead, Lulia, Ainscough; Jeffries, Gale; Hargreaves, Diskin, Kopczak, Elima, Bateman, Langley.
Replacements: L'Estrange, Addy, Manuokafoa, Burgess.
Referee: James Child (Dewsbury).