The Aviva Premiership pacesetters ran in eight tries to secure a winning bonus point.
Sam Smith scored a brace with Nick Easter, Matt Hopper, Rob Buchanan and Ben Botica also among the scorers for Connor O'Shea's high fliers, who also managed two scrum penalty tries.
The most Zebre could manage on a bitterly cold day was a first-half breakaway try by Leonardo Sarto and a late consolation for Alberto Chillon.
With Biarritz losing to Connacht last night, Quins' destiny now lies in their own hands and victory in next week's return against Zebre will effectively seal their passage through to the quarter-finals.
Harlequins were good value for their win and barely had to break sweat for some of their tries.
The first came for Smith on 26 minutes, a routine dash for the corner after the Zebre defence had been stretched on the back of a Maurie Fa'asavalu turnover.
Evans converted, adding to an earlier penalty, and already it looked as though the game was up for a desperately limited Zebre side, who failed to get out of their own half in the opening half hour.
Things went from bad to worse for the hapless Italians when serial offender Mauro Bergamasco was sent to the sin-bin for a ruck offence.
With that it looked as though the floodgates would open, and Smith helped himself to a second try after linking superbly with Tom Williams.
Again Evans converted to leave Quins 17-0 up and coasting.
Complacency cost Quins as they went in search of a third try before half-time, and it was Zebre who instead struck next, Sarto racing away after one too many passes cost Quins.
It was shortlived joy for Zebre, and Quins were back in business within three minutes of the restart as Easter bagged their third try after the visiting forwards wrestled the home pack into submission.
Danny Care, Matt Hopper and Mike Brown all went close to adding a fourth, but in the end it was the forwards that took the plaudits, forcing a penalty try after dominating the home scrum.
They would do so again later in the half as Zebre went into meltdown, and sandwiched in between was a fine individual try by Hopper, who showed great footwork to outmanoeuvre the home defence.
He was eclipsed shortly after, however, by substitute prop Buchanan, who scored with practically his first touch after another irresistible attack by rampant Quins.
A late try by Chillon proved scant consolation for a disillusioned Zebre, and it was left to another substitute, Botica, to wrap up the scoring for Quins.
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