The defender, who was included in a provisional Netherlands squad for Euro 2012 but did not make the final 23, arrives at Old Trafford as competition for established first choice Patrice Evra.
United allowed Brazilian full-back Fabio to join QPR on loan for the season and found themselves light in the position.
Sir Alex Ferguson said: "Alexander is one of the best young left-backs in Europe and we're delighted to sign him. He's someone we've been watching for a while now. He gives us some really exciting options in that position."
Buttner made 119 appearances for Arnhem after making his debut in the 2007-08 season, scoring 10 times.
He said: "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 team-mates and play my part in helping the club be successful."