The Bulls have to travel to Pontypridd, but will be full of confidence after they beat Division front-runners Bonymaen 26-22 at the weekend, courtesy of six goals from outside-half Rob Lloyd.
Team manager Justin Lewis said the hard-earned victory was perfect preparation for the trip to Sardis Road.
"We were 19-0 down after only 10 minutes, but we had chosen to play into the wind in the first half and that paid off," he said.
"It wasn't a day for the backs, so in the second half, we knew we had to keep it in the forwards and keep driving it up the middle. We've missed Rob Lloyd for the last three games, because he's a Marine and he's been away, and his kicking was crucial.
"We said before the game that we could win this one, and we knew it would give us a lot of confidence ahead of playing a Premiership side at Sardis Road.
"This season is strange - we can produce this sort of performance against the good sides, but can't seem to do it against the poorer sides."
In the other quarter-finals, Cardiff host Llandovery at the Arms Park, Maesteg take on Newport and Swansea welcome neighbours Neath to St Helen's.
Cardiff are staring Premiership relegation in the face, but will fancy their chances against the Drovers, who are only three points ahead on the table, having played one more game.
The two sides have met twice in the League this season, with results going to the home side. Cardiff won 21-13 in October, with Llandovery avenging that result with a thrilling come-from-behind 39-31 victory at the end of January.
Newport beat Maesteg 23-14 at Rodney Parade in October, but they will have to be on top of their game for the away trip this weekend.
Premiership table-toppers Neath will be favourites in their match against Swansea, having beaten the All Whites 17-6 in dire conditions last weekend. The last time Neath travelled to St Helens - in November - they came out on top 39-3.
Llandovery are aiming to make it three titles in a row in this year's Foster's National 7s being held at BT Sport Cardiff Arms Park on Sunday 28 August. Drovers assistant coach Simon Davies and Lee Rees assess their sides chances
A pioneering strength and conditioning programme has been delivered to more than 8000 boys and girls at hub schools throughout Wales this season, helping to develop stronger, more flexible athletes for rugby and other sports.
The Singha Premiership Rugby 7s Series returns to the BT Sport Cardiff Arms Park on July 22nd when all four Welsh regions go head to head. Scarlets skills coach Dai Flanagan and Blues Elite Performance Pathway Manager Richard Hodges along with current Welsh squad members were at the launch of this season's event
GB7s women left their England base to train in Wales this week. Olympic Games hopeful Rachael Burford gives WRU TV's Graeme Gillespie her thoughts on Welsh teammate Jasmine Joyce - a 'one touch wonder'
The GB7s women's squad is currently training at Wales' Centre of Excellence where assistant coach Richie Pugh tells WRU TV's Graeme Gillespie some 'tough calls' will have to be made when it comes to finalising the 12 spots for Rio's Olympic Games.