Van Gaal, 62, has signed a three-year contract to succeed David Moyes who was sacked in April after a disastrous run of results.
His duties with the Dutch national side means van Gaal will not be able to join up with United until after the World Cup in Brazil, which would not be until mid-July if the Netherlands reach the latter stages of the tournament.
Van Gaal, a disciplinarian who has won domestic titles and cups with Ajax, Barcelona and Bayern Munich as well as the Champions League, will become the first manager from outside the British Isles to take charge of United.
"It was always a wish for me to work in the Premier League," van Gaal explained.
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"To work as a manager for Manchester United, the biggest club in the world, makes me very proud.
"I have managed in games at Old Trafford before and know what an incredible arena Old Trafford is and how passionate and knowledgeable the fans are.
"This club has big ambitions. I too have big ambitions. Together I'm sure we will make history."
Giggs, United's record appearance holder who was in charge for the final four games of the season after Moyes' departure, added: "I am thrilled to have the chance to serve as assistant manager.
"Manchester United has been a huge part of my life and I'm delighted to be able to continue that relationship in such a key role," Giggs added.