Ademola Lookman and Zaniolo scored for the 2-0 win on Wednesday night.
Zaniolo later said: “I'm happy, it was a very difficult pitch. Stuttgart is very strong and winning here was far from a given. They beat Juventus and played against Real Madrid. We were good at defending and especially attacking.
“The goal gives a striker drive and confidence. I know I still have a lot to improve and understand the mechanisms but I'm giving it my all.
“When you play at high levels, there are expectations and you have to learn and live with them. I only think about training and giving my best. There are good and bad moments. I continue to follow my path, that of training and of the life of an athlete. Then on the pitch, if there is an answer, even better."
Asked about the difference between the Premier League and Serie A, the former Aston Villa loanee added: “Each team has its own characteristics and its own way of expressing itself. The Premier League is physically demanding, while in Italy there is more tactics and positioning.
"The difference. Maybe in Europe there is more space and in Italy less. In Europe there are no easy matches. But the real difference is in space and physical level."