The 24-year-old, who stole the show in last season's Europa League with a top-scoring 10 goals, has signed a four-year contract.
With his age and background, Kozak fits the profile of player Villa manager Paul Lambert likes to buy, with the Scot admitting to having tracked him for some time.
"We've been on this one for the last few months, trying to bring him to the club, so I'm delighted to get it across the line," Lambert told Villa's official website.
"Libor has an unbelievable scoring record in the Europa League, incredible really.
"He wasn't given much opportunity for Lazio in Serie A but he showed proper potential in leading the line there at a strong club with a great tradition and we need somebody like him as we only have Christian Benteke really as a main target man."
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."
Source: PA
Source: PA