The former Lazio striker has signed a four-year deal with Villa.
Kozak has won four caps for his national side and scored one goal in the friendly with Hungary last month.
"I'm really happy to be at Aston Villa and I'm ready for this new step in my career," he said.
"I hope it's a good step for me and also for Villa because everything is about the team. This is a big club, a very historic club with a fantastic fanbase and, for me, this is really important.
"From when I was a young boy, I have dreamt of playing in the Premier League and to do this with a club such as Villa is incredibly motivating. It's real now and I'm very happy about this."
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