The 23-year-old, who was part of the preliminary Dutch squad for Euro 2012, has made 119 appearances over the last four years at the Eredivisie club.
"Alexander is one of the best young left backs in Europe and we're delighted to sign him," United manager Sir Alex Ferguson told the club's official website.
"He's someone we've been watching for a while now. He gives us some really exciting options in that position."
Buttner, who becomes United's second Dutch signing in a week following that of striker Robin van Persie from Arsenal, added: "Joining Manchester United is one of the best days of my life.
"It's a team I have admired for many years now and I can't wait to join my teammates and play my part in helping the club be successful."