Zaniolo scored the second Giallorossi goal of the game to give his side the lead - a lead they would not relinquish, as Borja Mayoral added a late clincher to follow on from Edin Dzeko's equaliser.
Speaking afterwards, Zaniolo - who was a second half substitute - underlined his growing confidence heading into the new campaign.
"We are trying to do what the coach asks of us," he said.
"Today, in the second half, it was hard to press high up as it was so hot. But overall we are happy, we know we need to continue doing what we are doing right now. Next up it is Real Betis; we want to take things game-by-game and we hope to win as many of them as possible."
The Italy international was asked how training camp has been going for him so far.
"Really good," he said.
"We are training well, in good spirits and with that desire to get back out there every day. I think that's the most important thing. We want to be ready for the first game of the season."
On his own personal development, he added: "I feeling good, I'm feeling increasingly good about what I'm capable of. This is my 25th day working with the squad after a long time out, so I'm happy about that.
"I'm happy, it's a good group and I hope to be able to make my contribution to the team this year. Because I think we'd both benefit from that."
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