The 48-year-old was signed last year to coach the Ajax reserve team who compete in the second tier of Dutch football and finished the season as runners-up.
Keizer has been handed a two-year contract with the senior squad.
He coached previously at SC Cambuur, where he also spent most of a playing career that started with four appearances for Ajax’s first team in the 1987-88 season.
Bosz departed after one season in charge to take over at Germany’s Borussia Dortmund having taken Ajax to the Europa League final, which they lost 2-0 to Manchester United, and second place in the Dutch league behind Feyenoord.