The Aviva Premiership champions were never threatened by their Italian opponents and scored a total of seven tries to complete a crushing win.
Pool 3 leaders Quins can still theoretically be caught by Biarritz but need just one point from their remaining two games to secure a place in the knockout stages.
Quins were ahead within three minutes with a simple penalty from Nick Evans, who would kick a total of 18 points, before the outside-half set up the first try for Tom Casson with only seven minutes gone.
The hosts were then frustrated for the next 20 minutes due to some stout defence from the visitors, who ended up collecting the next score as Luciano Orquera burst through the defence before being hauled down, with Ruggero Trevisan on hand to pick up and score.
Evans responded with a penalty as Zebre, who had Daniel Halangahu yellow carded for diving in at the tackle area, before Quins were awarded a penalty try as the visiting back careered backwards at a scrum.
Evans converted to give the hosts a 20-5 interval lead, which was extended early in the second half when Ben Botica scored Quins' third try soon after coming off the bench, with Evans again converting.
Zebre then had another man sin binned as centre Gonzalo Garcia was penalised for deliberate offside and once again Quins took full advantage as a driving line-out was brought down just short, allowing Tom Guest to score.
The floodgates were well and truly opened by now and Sam Smith finished off a flowing three-quarter move for the fifth try and with 20 minutes remaining.
Mike Brown was replaced by Karl Dickson which resulted in Danny Care taking up an unaccustomed position. It paid dividends as the England international was soon presented with a 40-metre run in for another try.
With seven minutes remaining, Zebre's woes continued when flanker Filippo Ferrarini became their third visitor to the sin-bin before Dickson had the final say with the last try for the home side.
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