A deal was already in place for United to sign the 22-year-old for a fixed fee of £80.6million, with a further £4.2m in add-ons, to become the fourth most expensive signing in Premier League history.
With medical and personal terms now finalised, Antony joins Tyrell Malacia, Lisandro Martinez, Christian Eriksen and Casemiro in moving to Old Trafford this summer.
"This is an incredible moment in my career to be joining one of the most iconic clubs in the world," he said on United's official website.
Antony scored 25 times in 82 appearances at Ajax and is now looking forward to a reunion with Erik ten Hag, who left the Dutch giants over the summer to take charge of United.
"Playing under Erik ten Hag at Ajax was perfect for me and my development," added Antony.
"His style of football and coaching brings out the best in me, and I am excited by what he has told me about his plans and ambitions in Manchester.
"My time at Ajax was fantastic and I will always be grateful for the trust they put in me, but now I am ready for the next challenge, and I can't wait to join my new team-mates and play my part in bringing success to Manchester United."
Antony, who has been capped nine times by Brazil, is United's second most expensive signing - after the £89m paid for Paul Pogba in 2016 - and may not be the only arrival at Old Trafford on deadline day.
Ten Hag is also on the lookout for a supplementary goalkeeper for David de Gea, and Newcastle's Martin Dubravka has been strongly linked with a switch.