Keizer, 48, was trusted with Jong Ajax, the reserve team, last season and guided them to the runner-up spot in Dutch football's second tier.
Following Bosz's move to Dortmund shortly after leading Ajax to the Europa League final in his year in charge, Keizer has been trusted with the first-team.
The former Cambuur coach signed a two-year deal with the Amsterdam side with an option for a third.
"The conversations we had were really positive and I also feel trust from the club," Keizer told Ajax's official website.
"I was setting out to lead Jong Ajax to another good season in the Jupiler League, but in football things can suddenly change. This is obviously a very good opportunity that I will grab with both hands."
Edwin van der Sar, the club's CEO, says they are confident they have the right candidate.
"From the moment it seemed we would have to look for a new coach, Keizer was the choice of the technical heart of the club.
"He knows the players, the organisation and supports the philosophy of the club. Moreover, he showed last season with a clear style of play and performed excellently."
A product of Ajax's youth system, Keizer was a team-mate of current Ajax assistant coach Dennis Bergkamp among many other national stars, but played just four games.
As well as being a Cambuur player, technical director and coach, he also had spells in charge of Telstar and Emmen before returning to Amsterdam.