Bristol's Challenge Cup campaign ended with a 29-18 defeat at home by French side Narbonne on Sunday.
The result means Connacht will go through to the last eight as one of the best runners-up and will travel to Newcastle to face the Falcons.
In the other matches Gloucester will host Brive, London Irish will welcome Bayonne to the Madejski Stadium and Worcester will visit Northampton.
The games will be played between Friday, 31 March and Sunday, 2 April.
Bristol had to win to reach the last eight as one of the best runners-up but despite leading twice their weakened side was not up to the task against Narbonne.
Wing Sean Marsden gave Bristol the perfect start with an interception try in the eighth minute before Tommy Hayes extended that lead with a conversion and two penalties.
But Christophe Ruiz hit back for Narbonne with a converted try before combining with Frederic Lartigue to send Julien Pathe over five minutes later.
Marsden again gave Bristol hope when he finished off a blistering attack on 58 minutes, but English lock Andrew Springgay replied for Narbonne six minutes later.
Julien Candelon crossed in the final minute to bag a bonus point for Narbonne.
Gloucester booked their place in the quarter-finals thanks to a crucial 32-19 victory over pool rivals Bayonne on Saturday.
Prop Gary Powell crossed twice while Mike Tindall's try on the hour mark secured the bonus point as Gloucester romped home in front 8,000 fans.
London Irish made sure of a home quarter-final thanks to 75-12 thrashing of Section Paloise from Pau.
The Exiles scored 11 tries in total with Topsy Ojo scoring a hat-trick.
Newcastle, who like Irish had already reached the last eight, had Micky Ward to thank as they edged past Brive 24-17 at the Parc Municipal de Brive.
Ward popped up on the wing for the match-winning try to give the Falcons home advantage in the quarters.
"We had a few problems at the breakdown and we were frustrated at half-time, but I think we sorted things out in the second spell," said rugby director Rob Andrew.
Scotland's sole representatives Borders ended their campaign with an entertaining 39-23 win over pool four's bottom side L'Aquila in front of only 900 fans.
Tonga's Johnny Tuamoheloa made up for an earlier error to score the try that booked Worcester's place in the quarters.
Tuamoheloa's first-half sin-binning saw Montpellier go ahead 18-10 but his try helped the Warriors to a 31-21 victory.
Shane Drahm, Saosi Vaili and Thomas Lombard also scored a try apiece for the visitors.
Pool five rivals Connacht succumbed to a 28-24 away defeat against Catania.
The Irish side had gone ahead courtesy of tries from Ted Robinson before Catania's Barry Irving kicked the home team back into contention.
He was ably assisted by two Mauro Commuzzi tries before Matt Lacey took Connacht just one point shy with the scoreline at 20-19.
But Alejandrio Krancz ensured an Italian win when he pounced off the back row.