But the Italians replied almost immediately, Filippo Cristiano adding a second try before Kameli Ratuvou added the third late on for what turned out to be a comfortable win.
Edinburgh had lost on their last two visits to Italy but got off to the perfect start, snatching possession from the kick-off before Carl Bezuidenhout sent over a penalty for his side's first points of the afternoon.
Zebre came back immediately, getting close to their opponents' line before Samuela Vunisa knocked on, but with play taken back for an earlier Edinburgh infringement, Luciano Orquera levelled the scores.
Both sides were looking to attack but with the Zebre defence holding firm, Bezuidenhout went for a drop goal attempt, but it was charged down by Dries Van Schalkwyk with Orquera knocking on when looking to break clear.
But there was no let off for Edinburgh on 16 minutes as a series of missed tackles saw the Italians make ground down the blindside before Venditti crashed over in the far corner, Orquera hitting the post with his conversion.
Despite a period of Edinburgh possession in Zebre's half, it was the Italians who added to their tally on 28 minutes as Gonzarlo Garcia's long-range penalty tumbled over for 11-3 before Bezuidenhout missed with his own attempt for Edinburgh.
But the South African made amends on 31 minutes with his second penalty of the match for 11-6.
With the half-time break almost approaching, Scott put in a big hit on Garcia on the halfway line but after coming off his feet when trying to steal the ball, Garcia took the opportunity to add another three points for 14-6.
The interval seemed to galvanise the visitors however and they were back to within a point after five minutes as they pushed forward into their opponents 22, Tim Visser broke the line before feeding Scott for his third try in three games.
Cuthbert was successful with the conversion and while seeing his 48th-minute penalty drift wide, the Scottish outfit looked much improved.
But they were dealt a blow just minutes later as Zebre came again, Orquera's cute offload finding Quintin Geldenhuys who rammed his way towards the line, before Cristiano picked up the baton to cross. Orquera missed the conversion to leave the scores at 19-13.
Edinburgh saw Roddy Grant forced off injured on 57 minutes to be replaced by Sam Beard as the frantic early second-half action calmed somewhat.
And while the try line appeared temporarily out of reach for both sides, Edinburgh had a chance to reduce the difference from distance but Cuthbert failed to find the target from close to the halfway line.
Trailing by six with ten minutes to go, David Denton found himself in space for Edinburgh, but as had been the case of several occasions during the match, his offload failed to safely find the hands of his teammates.
And Zebre made the points safe three minutes from time, as Garcia claimed the ball from the lineout before feeding in replacement Ratuvou to cross under the posts, Orquera with the conversion to round off a successful day.
|Zebre Score Card|
|Edinburgh Rugby Score Card|