The defender, 22, is believed to have cost United £30m, and has signed a four-year contract with the option of another two years.
Bailly becomes United manager Jose Mourinho’s first signing at Old Trafford, with striker Zlatan Ibrahimovic tipped to join the club next month.
“It is a dream come true to be joining Manchester United,” he told mufc.com .
“To play football at the highest level is all I have ever wanted to do.
“I want to progress to be the best that I can be and I believe working with Jose Mourinho will help me develop in the right way and at the right club.
“I am looking forward to meeting my new team-mates and to starting this new chapter of my life.”
Bailly joined Villarreal last summer after a season-long stint at La Liga side Espanyol.
He made his Ivory Coast debut in 2015 and won the Africa Cup of Nations last year, scoring in The Elephants’ penalty shoot-out final win over Ghana.
“Eric is a young central defender with great natural talent,” Mourinho said.
“He has progressed well to date and has the potential to become one of the best around.
“We look forward to working with him to help nurture that raw talent and fulfil his potential.
“Eric is at the right club to continue his development.”
Bailly becomes only United’s second defensive signing from their past three transfer windows.
Right-back Matteo Darmian joined the club from Torino last summer, with Daley Blind the previous defender to arrive at Old Trafford back in September 2014.
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