After a series of trials, Alex got in.
It was his team's first match on Saturday - they won all their four games and Alex scored three times.
"He was pretty excited ... . he's a pretty international kid these days, but his first thoughts were, 'Maybe I could play for New Zealand.' That was his first reaction," Mr Cooke said.
When the family first moved to Milan, during the 2010 Fifa World Cup, Alex took great pride in wearing his All Whites shirt to school after the 1-1 draw against Italy.
And according to the academy's records, the Auckland boy is the first New Zealander on their books.
"We're pretty pleased, but more that he's just having a bit of fun. Give it three more years and we'll see if it eventuates into anything. As they get older, that's when you see whether there's any talent. But for now it's just about having fun," Mr Cooke said.
At the academy, Alex is already getting a taste of what a career in soccer would be like and gets treated like a professional in many ways, said Mr Cooke.
"At the start of a game they walk out to the pitch holding hands with music playing - it's quite spectacular."
Alex started playing at the Eastern Suburbs soccer club in St Heliers as soon as he was able to. He got the inspiration from his dad, who has played soccer all his life.
"So he comes from kind of a football-mad environment and he's come to the right place here. He gets to rub shoulders with [Sweden and Inter Milan star Zlatan] Ibrahimovic and a whole lot of other internationally renowned, world-class players and their kids.
"And I get to be the lucky dad who picks him up from school and stands next to superstar All Black-equivalents of the football world and chat to them about the weather - it's quite fun."
Mr Cooke said Alex had a natural intelligence for the game and, as well as being talented, he was also tall for his age which added to his abilities.
Mr Cooke said his son was devoted to the sport and he idolised Ibrahimovic as well as England's Scott Parker and Wayne Rooney.
Inter Milan
* Football Club Internazionale Milano, known as Inter Milan outside Italy, is a professional football club based in Milan, Lombardy.
* Established in 1929, the club has always been in the top-flight First Division. Inter have won 30 domestic trophies, the Coppa Italia seven times, and the Supercoppa Italiana five times.