Dean Richards' men arrived at Ravenhill with a 100-percent record to their name but they left without even a losing bonus point as Rory Best's late try gave Ulster a deserved 21-10 win.
Ulster, who went into the game in confident mood following a comprehensive Magners League win over Munster at Thomond Park on January 3, opened up a 16-0 first-half lead before defending heroically in the second period.
Conditions dictated the nature of the nature of the game in Belfast as a strong wind and driving rain made handling difficult throughout.
With the wind at their backs for the first 40 minutes, the home side got off to the perfect start as they scored the game's opening try after just 59 seconds. Poor Quins defending allowed outside centre Darren Cave to cruise through a midfield gap before racing unopposed to the try line.
Ian Humphreys added the extras before slotting three difficult penalties to move the 1999 champions two scores clear at the interval.
Fresh from stunning back-to-back victories against Stade, Quins started the second-half brightly as they closed the gap to just six points thanks to an early penalty and a 60th-minute penalty try.
It was all the visitors could muster, however, as Best's late score confirmed a hard-fought win for the home side.
Any feelings of disappointment were replaced by relief just over 24 hours later, however, with the Scarlets doing Quins the biggest of favours with a 31-17 triumph over the French aristocrats in West Wales.
If Quins arrived at the quarter final stages a week early due to the efforts of others, the Blues had only themselves to thank for their own qualification.
Despite being down to 14 men for 65 minutes of a fiercely-contested Anglo-Welsh encounter with Gloucester, the Blues made it five wins from five pool fixtures courtesy of a late Bradley Davies try.
Such an outcome seemed highly unlikely when Blues wing Tom James was sent off for alleged illegal use of the head with just 25 minutes played in the West Country.
James was dismissed after an altercation with Gloucester's French hooker Olivier Azam at a point when his side were already trailing 6-3 courtesy of two Olly Barkley penalties.
Barkley made it 9-3 with half an hour played and it stayed that way until two minutes into the second period when Leigh Halfpenny kicked his second successful penalty to cut the gap to three points.
Barkley re-established Gloucester's six-point advantage with 20 minutes remaining but the Cherry and Whites failed to pull away as Dai Young's men continued to dig deep in demanding conditions.
Having missed two earlier attempts at goal, Halfpenny's third penalty cut the deficit to just three points after 71 minutes before Davies stole the show with time running out.
Halfpenny kept his cool to add a vital conversion that left Gloucester chasing a try with just four minutes left to play and the heroic Blues defence did the rest to claim a quarter-final place for the second year running.