The Red Devils initially agreed a deal for the 17-year-old in May and have now finalised the deal, which will see the shot-stopper loaned back to the Serbian side for the 2014-15 season.
Milinkovic is an under-19 international and, upon his belated arrival in Manchester, 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," FK Vojvodina president Zoran Scepanovic stated.
"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. I want to point out Manchester United's very correct attitude [during the transfer talks]."