United landed Malacia in a reported £12.95 million ($15.4 million) deal after making a late move to sign the Netherlands full-back, who had been close to joining French club Lyon.
The 22-year-old has agreed a four-year contract with the option of a further 12 months.
It is a landmark deal for Ten Hag, who had been linked with a host of potential signings before finally making Malacia the first acquisition of his United reign.
Ten Hag arrived at United in May after finishing his successful spell at Ajax, where he had plenty of opportunities to see Malacia’s quality with Dutch rivals Feyenoord.
Malacia, who made 136 appearances for Feyenoord, said: “It’s an incredible feeling to have joined Manchester United. This is a new chapter for me, a new league with new team-mates and a tremendous manager leading us.
“I know from playing against his teams in the Eredivisie, the qualities that he has and what he demands of his players.
“Whilst I know that I am still young and will continue to develop, I can promise the United fans that I will leave everything on the pitch every time I pull on the red shirt.”
Malacia, who plays largely at left-back, is expected to be part of the United squad that leaves for the pre-season tour of Thailand and Australia on Friday.