Bath set up a quarter-final at home to Northampton with a 31-22 victory over fellow Prem side Saracens.
But Bristol are out in the last-16 with a 59-26 defeat against Toulouse.
Glasgow will host Toulon next, having got past Pretoria Bulls 25-21.
Leicester have a tough away trip to title holders Bordeaux this afternoon, while Edinburgh are at Leinster.