Bosz was last at Olympique Lyonnais in France, where he was fired last October, and before that had coached Ajax, Borussia Dortmund and Bayer Leverkusen.
The 59-year-old former Dutch international succeeds Van Nistelrooy, who made a dramatic departure from PSV last month with one league match left in the season after media revelations of a breakdown in his relationship with the board and key players.
A statement from the club on Monday said PSV and Bosz had extensive discussions about the vision, playing style and player group.
"I think my way of playing can lead to success. PSV has had periods in the past when they dominated Dutch football. I want to revive those times," it quoted him as saying.
Under Van Nistelrooy, they finished runners-up to Feyenoord in the Dutch league to qualify for the early round of the Champions League, and they won the Dutch Cup.