With only a home game against Connacht and a trip to French minnows Montpellier left in the pool stages, Bath's unbeaten record looks in little danger.
Bath full-back Nick Abendanon scored a crucial brace of tries in last weekend's 24-18 victory at The Stoop, and was over the line in only the second minute of the Pool Four return match.
Bath's Olly Barkley and Quins' Adrian Jarvis traded penalty goals, but Shaun Berne's try in the final minute of the first half gave the home side a daunting 17-3 lead at the break.
Jarvis pegged back three points early in the second spell, but an upset was never on the cards as Bath closed ranks. An injury-time try to winger Tom Williams and a penalty goal from former All Black Andrew Mehrtens gave Quins the final say and a losing bonus point.
Saracens took control of Pool Two with a 36-16 win over GRAN Parma in Italy.
After hammering Parma 71-16 last weekend, the only surprise about Saracens' first try was that it took 18 minutes to arrive.
As scrum-half Alan Dickens limped off after taking a heavy knock, the Saracens pack drove their rivals backwards and French referee Romain Poite immediately awarded a penalty try.
The two New Zealand-born fly-halves - Parma's Rima Wakarua and Saracens' Glen Jackson - exchanged penalty goals as rain started falling at Stadio Sergio Lanfranchi, before centre Rodd Penney's try gave the visitors a 17-9 lead at the break.
Flanker David Seymour finished off a brilliant break from Thomas Castaignede to put the result beyond doubt, while there were also tries for Paul Gustard and replacement scrum-half Moses Rauluni.
In Romania, Newport Gwent Dragons kept the pressure on Pool One leaders Bristol with a 39-29 win over Bucuresti.
Home winger Catalin Fercu scored two first-half tries, but his one-man show could not prevent Turner's men from recording a crucial bonus-point victory.
Turner said: "We're very disappointed. Every time we scored, we leaked a try. We've won with five points, which is what we set out to do, but we never do anything easy.
"We've got a few bruised egos today, and I think the players will have more respect for Romanian rugby after this. Bucuresti gave us some game."
Both sides scored five tries, but 14 points from the boot of Wales international Ceri Sweeney proved the difference.