The Red Devils initially agreed a deal for Milinkovic in May and have now finalised the deal, before loaning the teenager back to Vojvodina.
The Serbian teenager is an Under-19 international and, upon completing his loan spell at his current club, will compete with David De Gea, Anders Lindegaard and Ben Amos for a starting spot at Old Trafford.
"After lengthy talks that have taken place over the past month, we have finally completed negotiations on Monday and we can officially inform the public that our young and promising goalkeeper has joined Manchester United," said the Serbian club's president Zoran Scepanovic.
"He will remain in Vojvodina for the next year, where he will be under the supervision of our experts and coaches from the English football giants, while continuing to train and perform in the jersey of his home club.
"This is an opportunity to thank everyone who participated in the realisation of this transfer. Can I point out Manchester United's very correct attitude."