The 18-year-old had already agreed to join the Eredivisie side at the end of the season.
However, a deal has now been struck to allow the Cameroonian to immediately make the switch from Camp Nou to the Amsterdam Arena.
La Liga side Barca had previously hit out at Ajax for their handling of the initial deal for Onana, with since deposed director of football Andoni Zubizarreta slamming the Dutch club for "disrespectful" behaviour.
"I am very glad I have made the move to Ajax," Onana told his new club's official website.
"There were quite a few clubs interested but I honestly did not have to think about it.
"When Ajax showed interest it was immediately clear to me that I wanted to go to this club."
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