Kozak, who was the top scorer in the Europa League last season, has signed a four-year contract with the Premier League side.
Villa boss Paul Lambert said: "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.
"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.
"He'll take that burden off Christian and he'll add greatly to what we've got, with the likes of Nicklas Helenius, who's a different type of player, Gabby Agbonlahor, Aleks Tonev and Andi Weimann in that area of the pitch where we can hurt teams.
"Libor's powerful, too, and I'm looking forward to working with him. We needed to bring in someone like Libor to help the overall philosophy to work."
Kozak said: "I'm really happy to be at Aston Villa and I'm ready for this new step in my career.
"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 fan base 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."