The 39-year-old, who won 42 caps, has signed a contract until June 2016 to succeed Mirko Slomka, who was sacked last week.
Korkut, who was born in Stuttgart, most recently worked as a coach with the Turkey national team.
Hannover sporting director Dirk Dufner told the club's official website: "This appointment is a conscious decision for the future of Hannover. With Tayfun Korkut we have chosen a young, highly-qualified and highly-motivated coach.
"We know that we have made a brave decision, but see great prospects in the partnership with Tayfun Korkut. I have great confidence that we will be successful together."
Korkut added: "I expressly thank Hannover for their trust in giving me this chance."
Korkut, who speaks German, will take his first training session on Sunday following the team's winter break.
He started his playing career with Stuttgart Kickers before spending five years at Fenerbahce and also played in Spain for Real Sociedad and Espanyol.
He has previously coached age-group teams at Sociedad and, in the Bundesliga, at Hoffenheim and Stuttgart.