The Dutch champions have signed a two-year deal with the 49-year-old, who is on the brink of taking Club Brugge to the Belgian title.
Schreuder was an assistant to Ten Hag at Ajax for 18 months and with Ronald Koeman at Barcelona last season.
They have come from behind to lead the Belgian league in the closing stages of the season and will seal the title if they win on Sunday.
"Becoming head coach at Ajax is an honour and a great opportunity for me, which I grab with both hands," he said in a statement on the Ajax website.
"I would like to win prizes in Amsterdam with a team that plays attractive football. With a good balance between experienced top players and many great talents.
"But first I want to finish the season successfully with Club Brugge. I will do everything I can to make that happen."