Two tries from Danny Care set Quins on their way to a deserved success at the Sportsground in Galway.
Eric Elwood's men ruined Quins' hopes of reaching the knockout stages last January with a nail-biting 9-8 defeat at the Galway venue.
Nine months on, both sides threw off the shackles to share out 43 points in a free-flowing first half full of enterprising rugby.
Dan Parks converted David McSharry's seventh-minute try and added four penalties to move the hosts 19-9 ahead.
But Quins grabbed 15 points in the closing 11 minutes before the break, that profitable spell being crowned by Care's two shrewdly-taken tries.
Ben Botica landed an injury-time penalty to complete a 24-19 turnaround by the interval.
Botica turned in another dazzling place-kicking display with a 20-point haul, as two further three-pointers from him killed off Connacht's doughty challenge in the Pool Three clash.
Saracens' dominant first half ensured their experiment of staging a home European match overseas was rewarded with a 30-13 Pool One victory over Racing Metro at King Baudouin Stadium in Brussels.
They ensured their journey was worthwhile as they established a 20-3 interval lead that Racing, with former England back Olly Barkley pulling the strings at 10, were unable to overhaul.
Tries from Chris Wyles and Steve Borthwick put the 2011 Aviva Premiership champions in the driving seat, but their opponents were only fully subdued by Will Fraser's touchdown four minutes from time
Clermont Auvergne, last season's semi-finalists, went back to the top of Pool Five thanks to a 13-minute burst at the start of the second half on the way to a 46-12 victory over Exeter.
The visitors were quick off the mark at the restart, scoring 15 points to regain the lead - which they held on to and increased without reply at Sandy Park.
Elsewhere on Saturday, Leinster went to the top of Pool Five with a 20-13 win away to the Scarlets.
Wing Isa Nacewa scored the visitors' only try at Parc y Scarlets with Jonathan Sexton supplying 15 points with the boot.
The Scarlets, who came away with a losing bonus point, scored a try through Gareth Maule and two penalties and a conversion from Rhys Priestland.
It gave Leinster their 17th successive victory in the tournament and coach Joe Schmidt's 17th win from 19 Heineken Cup games.
Toulouse narrowly avoided a shock against Treviso as the French side hit back to emerge 33-21 winners in Pool Two.
Treviso, backed by a passionate crowd at the Stadio Comunale di Monigo, led 18-9 at the break thanks to six penalties from fly-half Kristopher Burton.
Toulouse's Luke McAlister and Burton exchanged penalties after the break before both the visitors' Vincent Clerc and Treviso's Robert Barbieri were sin-binned.
Toulouse got themselves back in the match, though, courtesy of a penalty try and then moved ahead when Louis Picamoles burst through to go over.
Treviso were tiring and the visitors extended their lead when McAlister chipped the ball on for Clerc to touch down under the posts.
Finally, in Pool Three, Biarritz grabbed their first win with a 38-17 romp at home to tournament newcomers Zebre.
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