Sarries' chances of automatic progression were ended by a 15-15 draw with the Ospreys last weekend, leaving them hoping to qualify as one of the three best runners-up.
The result in Swansea meant the Welsh region needed to win at Clermont Auvergne to be sure of second spot, but the Top 14 side triumphed 24-7 to win Pool 2 by four points from Sarries.
Later results could have ensured Mark McCall's men would finish among the top three second-placed sides, but a losing bonus point for Toulon in defeat to the Scarlets mean they could yet be eliminated.
Sarries ran in seven tries and benefited from 11 successful kicks from Owen Farrell as they piled more than 50 points on Northampton for the third time this season.
The Scarlets, meanwhile, turned in a determined performance to defeat Toulon 30-27, leapfrogging the three-time winner to top Pool 5.
The win earns the Scarlets a home quarter-final, while Toulon will be on the road in the last eight. As things stand, the teams will meet again in the next round.
Bath's 47-28 win at Benetton Treviso proved to be in vain as the two sides in action at Parc y Scarlets took the top two spots.
Premiership leader Exeter Chiefs crashed out of the tournament after a 28-21 defeat to Glasgow Warriors, which featured a stunning team try finished off by Tommy Seymour.
Leinster won 23-14 at Montpellier to finish with six wins from six in Pool 3 and a 12-point cushion over the Chiefs.