The 17-year-old has been linked with a move to the Eredivisie leaders for some time after breaking into the Groningen first team last season.
Last month director of football Marc Overmars confirmed Ajax were interested in signing Zivkovic and Groningen revealed talks were ongoing over his potential transfer.
Those talks have now been concluded and the forward, who has six league goals in 23 appearances this season, has signed a three-year contract at Ajax.
Zivkovic admitted signing for the 32-time league champions was a dream come true and revealed he had big a fan of the club since a young age.
"Ajax is a fantastic club with a fantastic stadium," he told the club's official website. "From early on I have been an Ajax fan.
"I have good competition here at Ajax, but I'm quite confident.
"I am someone who adapts quickly. I hope to play as much as possible."
Zivkovic will finish the current campaign at Groningen and will try and maintain their mid-table position in the final six matches.
