A late flourish by Toulon in their bonus-point 27-12 victory over Sale ensured that coach Mike Ford's side will travel to Saracens next week aiming for an away win to nail down one of the three best runners-up places for a spot in the knock-out phases.
In Bordeaux, France fly-half Camille Lopez kicked three penalties for Clermont in an error-strewn game in which they were reduced to 13 men early on.
Irish counterpart Ian Madigan booted two of his own penalties as the visitors ran out 9-6 victors.
Clermont, with 21 points in Pool 5, joined Irish provinces Munster and Leinster, from Pools 1 and 4 respectively, in the last eight.
"It's mission accomplished," said Clermont captain Damien Chouly. "We had our destiny in our hands and a victory here has opened up the doors for a quarter-final. We've done it but it wasn't easy."
There was no record 14th Champions Cup win in a row for Saracens, who are guaranteed at worst one of the three best runners-up spots on offer.
They had to rely on an injury-time try from England winger Chris Ashton, converted by Owen Farrell, to snatch a 22-22 draw in Llanelli.
"Owen was incredible in that last phase of play. He really set the tone, and made two or three half-breaks before he made the full break. It makes things exciting for next week because it's going to a big, big game" against Toulon at home, said Saracens coach Mark McCall.
"We will be playing to win the pool next week and more importantly for a home quarter-final. Home quarter-finals are a very good thing to have. It was more important to get a home tie than break the record for successive wins."
Scarlets had dominated much of the game and looked set for a memorable victory over the champions.
But a turnover at the death handed Saracens the opportunity they needed.
Winger Nathan Earle scored the English side's first try thanks to fine offloading skills from full-back Alex Lowzowski.
Dan Jones kept the penalties ticking over for a 9-5 halftime lead for Scarlets, stretching that with his fourth early in the second half.
But Lowzowski again turned provider, his break and inside pass seeing Ashton scoot through for his first try.
Scott Williams hit straight back, lining up a long, flat pass from scrum-half Aled Davies to cross under the posts.
Farrell and Jones traded penalties before the England fly-half made a crucial break and found the ubiquitous Ashton on his shoulder to break Scarlets' hearts.
Saracens, on 20 points, now top Pool 3, three-time champions Toulon sitting pretty in second after their win at home over Sale.
Jonathan Pelissie and Fiji winger Josua Tuisova scored a late brace of tries to guarantee a bonus point after Ma'a Nonu and Bryan Habana had earlier crossed the English club's whitewash.
There's still all to play for in Pool 2, headed by Connacht (18 pts) but with Wasps (17) in the driving seat with a final group match at hapless Zebre who have lost all five games, shipping 290 points.
Connacht have to travel to four-time champions Toulouse where both sides will still be in contention to make the last-eight.
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