Scores from Rodney Ah You, Kieran Marmion and a penalty try meant that the visitors went into the half-time break in Italy 22-0 up.
And things got better after the break when Eoin Griffin dotted down to seal a bonus point for the Irish region.
Zebre hit back through tries from Giovanbattista Venditti, Samuela Vunisa and Dries van Schalkwyk, but there was to be no way back.
And head coach Cavinato could take little solace after their slow start cost them dear, meaning Connacht have now beaten them four times this season.
"We were too tense at the start: when after five minutes you already 8-0 down it's difficult to turn it around mentally," he said.
"We tired ourselves out just by trying to find our lucidity. We knew that they were strong in the scrum, an area where we really had a hard time.
"In the first ten minutes we didn't defend their maul correctly and we didn't rush out aggressively in defence.
"At 15-0 I'd have preferred that we took the easy kick to get back to 15-3 instead of going down 22-0.
"In the changing rooms we spoke to each other and said we needed to play as we had spoken about in the week, taking the points on offer. That's what happened in the second half and we scored."
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