The much-travelled Daum replaces Anghel Iordanescu who stepped down after Romania's failure to make it past the first round of Euro 2016.
They finished bottom of Group A after losing to hosts France and Albania and drawing with Switzerland.
Romania will play Denmark, Poland, Montenegro, Armenia and Kazakhstan in World Cup Group E qualifying.
The Romanian Football Federation (FRF) said a number of top German coaches, including Joachim Loew, Juergen Klinsmann and Ottmar Hitzfeld, had sent messages in support of Daum's appointment.
"I was glad to hear that he can become coach," Germany boss Loew was quoted as saying. "It is a good decision for Christoph Daum and a good decision for Romanian football."
However, several home-grown coaches have criticised the appointment of the 62-year-old.
Daum becomes the second foreign coach of Romania. Austrian Josef Uridil led the team at the World Cup in 1934.
He has solid Bundesliga credentials. He steered VfB Stuttgart to the title in 1992 and has also coached Bayer Leverkusen, Cologne and Eintracht Frankfurt.
In addition Daum has won three league crowns in Turkey, two at Fenerbahce and one with Besiktas, and the Austrian championship with Austria Vienna in 2003.
Daum has been out of work since he was sacked by Turkish club Bursaspor in March 2014.
(Reporting by Angel Krasimirov; Editing by Tony Jimenez)
