Bath beat Montpellier 42-17 to take top spot in European Challenge Cup Pool Four and a place in the quarter-finals.
Harlequins failed to qualify despite clinching the runners-up spot behind Bath with a 26-10 win over Connacht.
Newcastle, who won 50-6 at Petrarca, progressed after finishing on the same points as Quins because they had scored more tries than their rivals.
Worcester triumphed 19-16 at Viadana in Pool Five to finish runners-up but they miss out on the quarter-finals.
Mike Stephenson scored twice while Olly Barkley, Michael Lipman, Nick Walshe and Shaun Berne also touched down as Bath cruised to victory.
Chris Hala'Ufia, Stuart Abbott, Hal Luscombe and Mike Brown notched Harlequins' tries and Adrian Jarvis added three conversions.
Connacht's try came from Conor O'Loughlin, with Peter Durcan converting and Mark McHugh kicking a penalty.
Ollie Phillips scored two touchdowns and Toby Flood also had a try and five conversions in Newcastle's emphatic win in Padova.
Worcester led 14-5 at half-time against Viadana but held on as Thinus Delport (2) and Aisea Havili got the tries.
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Newport Gwent Dragons booked top spot in Pool One after winning 29-15 in Bayonne on Friday.
Kevin Morgan's two tries helped the Dragons hit back from 8-3 down at half-time with Aled Brew and Andrew Hall also crossing.
Bristol finished as one of the best runners-up to qualify for the last eight after a 33-19 win over Bucuresti.
Glasgow reached the quarter-finals of a European tournament for the first time thanks to a 6-6 draw with Saracens.
Dan Parks matched the two penalties from Glen Jackson to help his Glasgow side qualify as one of the best runners-up.
Saracens remain unbeaten in the competition this season and have a home quarter-final after finishing top of Pool Two.
Bristol and the Dragons were locked on 20 points, but the Welsh region had a superior try-count over two games against the West Country side.
Bristol romped to victory thanks to tries from Mariano Sambucetti, Rob Higgitt, Craig Short with Craig Morgan scoring twice.
European Challenge Cup quarter-final line-up:
Matches to be played weekend of 30/31 March/1 April
Clermont Auvergne (No 1 seeds) v Newcastle (8)
Saracens (2) v Glasgow (7)
Bath (3) v Bristol (6)
Newport-Gwent Dragons (4) v Brive (5)