Tom Guest, who switched to number eight in the absent of the unwell Nick Easter, gave Quins an ideal start when he sliced through from 35 metres, and then made a key contribution to a second try for the team 10 minutes into the second half when he made a break before playing a long ball out for Mike Brown to supply the finish.
Miah Nikora kicked all of Connacht's points, whick Nick Evans booted two penalties and two conversions for Quins.
Leeds Carnegie 55-7 Rugby Parma
Marco Wentzel and Lee Blackett scored two tries apiece to help Leeds Carnegie record their second win of the season as they crushed Crociati 55-6.
The victory was Leeds' first win at Headingley since they retained their place in the Premiership with a narrow victory over Worcester Warriors in April.
Ceiron Thomas, Leigh Hinton, Pete Wackett and Henry Fa'afili also touched down as Leeds converted a 13-6 half-time lead into a far more comprehensive final scoreline.
Aviron Bayonnais 65-7 I Cavalieri Estra
Aviron Bayonnais destroyed Italian side Cavalieri in a one-sided encounter at Stade Jean Dauger.
The French side effectively secured the bonus point victory before half-time as they ran in the first four of their nine tries before the break.
South African centre Sam Gerber was the star for Bayonnais with four tries in a win that puts his side top of the Pool 1 on points from Harlequins, with Luca Petillo scoring a consolation effort for the Italians.
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Exeter Chiefs 36-10 Newcastle Falcons
Exeter kept alive their hopes of qualifying for the quarter-finals of the Amlin Challenge Cup as they picked up maximum points against Newcastle with a 36-10 win.
Even though coach Rob Baxter made 12 changes to the side that beat Wasps last weekend, Exeter ran in three tries before the break to open up a 22-0 advantage.
That left the Chiefs with the goal of claiming one more try in the second half to match the bonus point that pool leaders Montpellier had gained in their 39-14 home win over Bourgoin the night before.
They managed two more to clinch the full five points and are now only three points behind Montpellier at the top of the table, who they face in France in Round Six next month.
Ben Moon's try got things going, Michael Stupple added a second try and Ryan Davis kicked two conversions and a penalty as they assumed control.
Matt Jess went over for a third try, with the fourth and fifth scores coming from Chad Slade and Andrew Higgins.
Newcastle grabbed consolation tries from Mickey Youngs and Danny Williams.
La Rochelle 6-13 Gloucester
Gloucester closed the gap on Pool Six leaders La Rochelle with a hard-fought 13-6 victory to put their Amlin Challenge Cup hopes back on track.
James Simpson-Daniel crossed for the only try of a scrappy affair on 25 minutes in freezing conditions on the west coast of France.
The result leaves Bryan Redpath's men just one point behind their rivals at the top of the table.
Benjamin Dambielle slotted La Rochelle's points, with Nicky Robinson kicking a conversion and Freddie Burns two penalties to make up Gloucester's score.
Brive 18-9 Sale Sharks
Sale fell to a disappointing 18-9 defeat at the hands of Brive in Pool Two of the European Challenge Cup to compound a difficult week at the Edgeley Park club.
The Shrks showed plenty of attacking intent but were always on the back foot against a clinical Brive outfit after Alexis Palisson grabbed two tries to help propel the French side into an 18-3 lead after little more than half an hour.
Nick MacLeod cut the deficit with two more penalties, but the Sharks lacked the cutting edge needed to break down their opponents.
Both clubs had hit the headlines for all the wrong reasons in the build-up to the game with off-field problems but it was Brive that ended the week on a high.
Bucharest Oaks 20-29 Stade Français
Stade Francais came dangerously close to a shock defeat against Bucharest Oaks in Romania, with the home side still within striking distance of their illustrious visitors before Noel Oelschig's late penalty put the result beyond doubt.
The result puts Stade ten points clear at the top of Pool 4 with 14 points from four matches.
Petrarca Padova 37-10 Cetransa El Salvador
Two tries apiece from Martin Acuna and Zane Ansell helped power Padova to a thumping win over El Salvador in Pool 2 to give the Italian side their first points of the European Challenge Cup campaign.
Rugby Rovigo 10-33 Agen
Agen cruised to an easy victory over Rovigo in Italy, with winger Kevin Swiryn scoring two tries - the first after just five minutes of the match - to leave the French side joint top of Pool 5 with La Rochelle.