The Hungarian Football Association (HFA) agreed terms with Bozsik, 44, and his assistant coach, former Hungary midfielder Lajos Detari, on March 16 and they put pen to paper on Thursday.
The contract expires at the end of the 2008 European Championship qualifying campaign in September 2007 and will automatically be renewed should Hungary finish in the top three of their group.
Hungary have failed to qualify for a World Cup or a European Championship for more than 20 years.
Bozsik replaces former German international Lothar Matthaeus whose contract ran out at the end of 2005.
Bozsik, 44, has lived in the shadow of his famous father Jozsef, a member of Hungary's Golden Team, who died in 1978.
Bozsik coached Zalaegerszeg to their first championsip in 2002 when they also beat Manchester United in Budapest in the first leg of their Champions League third round qualifier.