The 53-year-old has signed a two-year contract that will keep him at Signal Iduna Park until June 30, 2019.
In his only season at Ajax, Bosz guided the Amsterdam giants to second place in the Eredivisie and the Europa League final, drawing praise for an exciting brand of football with a young group of players.
The former Vitesse and Maccabi Tel Aviv boss takes over from Thomas Tuchel, whose strained relationship with Dortmund chief executive Hans-Joachim Watzke ultimately cost him his job.
Dortmund were heavily linked with a move for Nice coach Lucien Favre, who built an impressive reputation for himself in Germany by turning Borussia Monchengladbach into consistent top-four challengers.
However, the Ligue 1 side issued a statement insisting the Swiss coach would not be allowed to leave Allianz Riviera.