That was enough to push them off the bottom of the table, leapfrogging Connacht, who they face this weekend in European competition.
In the return fixture Zebre were comfortably beaten 33-6 but Cavinato wants them to provide a much better showing after a disappointing European campaign to date.
He said: "This win [over Treviso] gives us a little tranquillity ahead of the game in Galway on Saturday. When the result goes against you, you're often aware of the good work but end up frustrated.
"Against Treviso that wasn't the case. We need to make up for the poor displays that we've shown in Europe this year. We need to play well in Europe starting this weekend against Connacht."
Hooker Davide Giazzon was the man of the match in the win over Treviso and called on the team to continue their good form in 2014.
He added: "It was a win that we'd been waiting for since last year, we weren't far away in the game at Treviso. This time we saw the Zebre that we want to see all year long, playing as a team and winning the game.
"Playing in the front row is a collective effort so while I'll gladly accept my first man of the match award, I think the entire Zebre scrum deserves it."
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