The Southampton hitman marked his Azzurri debut by netting a 24th-minute winner as Antonio Conte's side maintained their 100% start to Euro 2016 qualifying.
"I am happy, everyone has a different journey and I am glad I got here, especially as to score is the most important thing for a striker," he told reporters.
"I had another chance too and it's a shame that one didn't go in. On the one hand it's positive, but on the other I am very harsh on myself, as I could've done a lot better.
"I am Italian and so proud to be wearing the Azzurri jersey. I hope I get the opportunity to wear it again."
Italy face co-leaders in Group H Croatia on November 16.