Centre Christian Loamanu scored the only try of the game early in the second half, while Alberto Di Bernardo scored the remainder of his team's points, but his three penalty misses allowed the Zebre to stay in touch.
However Smith is adamant that against a strong Zebre defence, his team can be proud of their efforts ahead of the return fixture in Treviso on December 29.
He said: "They desperately didn't want to lose and it seemed like a game from a few years ago. And you have to remember how heavy the pitch was.
"I'm delighted with the win. The Zebre are a determined side, and they wanted their first win. "An Italian derby is always like that. We had 80 percent of possession, always attacking and they did everything to stop us scoring.
"I don't think you can be too negative about how we played, even if we probably should have scored more points. Our maul went well though, and the scrum as well. We controlled what we could but considering the grit of the opposition, they defended with heart and deserve credit for that."
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