The Quins took some time to subdue an enterprising Irish side and were fortunate to lead 6-3 after the first half an hour. Their set scrum then emphasised their superiority to ensure there was no way back for Connacht, who were briefly reduced to 12 men in the second half as they fell to a heavy defeat.
Tom Williams (two), Ugo Monye, George Lowe and Ben Botica touched down for Quins, who were also awarded a penalty try, with Nick Evans and Botica kicking the rest of their points.
Matthew Jarvis kicked an early penalty for Connacht, with John O'Connor crossing for their only try of the game in the second half.
Biarritz Olympique recorded a 32-6 bonus point victory over Zebre at Stadio XXV Aprile to keep alive their hopes of a quarter-final spot as one of the best runners-up. A brace of tries from full back Marcelo Bosch sandwiched scores from Benoit Baby and Seremaïa Burotu.
Toulon demolished Cardiff 45-25 to march into the knockout stages.
The Blues made a bright start at Stade Felix Mayol and surged into an 8-0 lead thanks to Leigh Halfpenny's early try.
But Toulon fought back and ran riot with seven tries through Rudi Wulf, Xavier Chiocci, Jean-Charles Orioli, Maxi Mermoz, Alexis Palisson, Mathieu Bastareaud and Florian Fresia.
The bonus-point victory sees Toulon restore their five-point lead at the top of Pool Six and while they are already through, they can secure a home quarter-final in next week's trip to Montpellier.
The Blues fought back with tries from Alex Cuthbert, Dafydd Hewitt and Halfpenny but it was a mere consolation.
Owen Farrell kicked a record 10 penalties to clinch a sensational 37-28 victory for Saracens at Racing Metro.
The England fly-half was imperious with the boot in the face a partisan French crowd with a perfect 11 kicks from as many attempts in a staggering 32-point haul.
Chris Wyles scored Saracens' only try of a tense clash though the Aviva Premiership side had to cope with two yellow cards, a try deficit of three to one and a worrying injury to David Strettle to pull off a vital victory.
The result leaves Saracens one win away from reaching the quarter-finals for the second year in succession.
Leinster rediscovered their try-scoring touch with a 33-14 bonus-point win over the Scarlets which keeps their hopes alive.
First-half touchdowns from Cian Healy, Shane Jennings and man-of-the-match Luke Fitzgerald saw Leinster sweep into a 19-6 lead at the RDS.
Scarlets full-back Liam Williams snuck in for a five-pointer before the break, only for his opposite number Rob Kearney to reply with the all-important bonus-point score.
Ian Madigan came off the bench to grab a fifth try in the final minute, as the Irish province's focus switches to a decisive clash at Exeter next week where they need another bonus point victory.
Clermont Auvergne ran in six tries to secure their place in the knockout stages with a 46-3 bonus-point victory over Exeter.
The French giants led 20-3 at the break thanks to tries from Wesley Fofana and Jean-Marcellin Buttin.
Exeter fought bravely following the restart, but the pressure eventually told and Clermont ruthlessly completed the job with further touchdowns for Damien Chouly, David Skrela, Buttin and Napolioni Nalaga.
The victory, Clermont's 54th in a row at Stade Marcel Michellin, secures their place in the quarter-finals as Pool Five winners and they can ensure a home draw in their final group-stage match against the Scarlets.