In October 2007, he scored his first hat-trick, coming in a 5–1 Opava win against FC Zenit Čáslav in the second division.

[7] At the end of his loan spell, Kozák returned to Lazio and scored his first goal for the Biancocelesti on 18 September 2010, which turned out to be the winner against Fiorentina.

[12] On 2 September 2013, Kozák signed a four-year contract with English side Aston Villa for a €6.5 million transfer fee,[11] being assigned the number 27 shirt.

[15] On 2 January 2014, Villa announced that Kozak had broken his right leg in a training ground collision with teammate Ciaran Clark, and would miss the rest of the 2013–14 season.

[17] He eventually made his comeback 15 months after the initial leg break in March 2015 in an under-21 match against Wolverhampton Wanderers.

[29] On 14 November 2012, Kozák debuted for the Czech Republic national football team in a friendly match against neighbours Slovakia, coming as a substitute for David Lafata in the 67th minute.