Czech Republic striker Kozak, 24, who netted 10 times during Lazio's run to the quarter-finals of Europe's second-tier competition last season, has agreed a four-year deal with Villa.
"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."
The 6-foot-4 inch Kozak, who has won four caps for the Czechs, is Villa's seventh capture of this transfer window.
Villa also announced on their website that former Ireland midfielder Ireland had joined Stoke on a season-long loan.
Stoke boss Hughes managed Ireland at Manchester City and the club's chief executive Tony Scholes hopes the midfielder can recapture the form he showed then.
Scholes said: "Stephen is a player that Mark knows really well and when the opportunity arose he was keen to add him to his midfield and attacking options.
"Arguably the most successful period of Stephen's career to date was when he played for Mark at Manchester City and we hope that he can rediscover that form."