The 23-year-old made his first appearance as Italy beat Netherlands 2-0 in Antonio Conte's first game as coach last week, before scoring the opener in his country's Euro 2016 qualifying opener - a 2-0 win in Norway.
And, after settling quickly into life as part of the national team, Zaza has no intention of relinquishing his place.
"It [the Norway match] was certainly a different game," he told Rai Sport. "Perhaps the Dutch were stronger in terms of quality, but Norway were very determined and we struggled for a while, but eventually achieved our targets.
"I am living a dream week. Now my objective is to stay in this squad and continue to do well.
"Conte's approach is very tiring, but if you want to do well and play as a team, then you have to run. That's how it should be."
Zaza will be back on Serie A duty this weekend as Sassuolo take on Inter.
