Slomka - who led Hannover into the Bundesliga and helped them qualify for the UEFA Europa League twice - parted company with the club on Friday.
And Hannover acted swiftly in appointing Korkut, who left his post as Turkey assistant coach to take on the job.
The 39-year-old has signed a contract until June 2016 and said on Tuesday: "I'd like to expressly thank Hannover 96 for the confidence they're placing in me by giving me this opportunity.
"The talks with (Hannover sporting director) Dirk Dufner were conducted on a basis of mutual trust and I'm really looking forward to the demanding but fascinating challenge lying ahead for us."
Dufner expects big things from Korkut and said: "This is a considered appointment with an eye to the future of Hannover 96.
"In Tayfun Korkut we have chosen a young, highly qualified and supremely motivated coach whose personality and track record utterly convinced us that he's the right man for the job."
Korkut won 42 caps for Turkey in a playing career that saw him spend time at Stuttgart Kickers, Fenerbahce, Beskitas, Real Sociedad and Espanyol.
Hannover sit 13th in the Bundesliga table after a run of just one win in 11 games, and are just four points above the bottom three.
