Van Gaal has signed a three-year deal and will take up his post when his duties as Netherlands manager end after this summer's World Cup.
Although the tournament in Brazil will now command Van Gaal's immediate intention, the 62-year-old Dutchman spoke of his determination to make an impact at United.
"It was always a wish for me to work in the Premier League. To work as a manager for Manchester United, the biggest club in the world, makes me very proud," said Van Gaal.
"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. I'm sure we will make history."
Following David Moyes's sacking in April, United quickly identified Van Gaal as their preferred successor with the former Ajax, Barcelona, AZ Alkmaar and Bayern Munich coach boasting a cv detailing league titles with each of his clubs as well as Champions League glory with Ajax.
Ed Woodward, United's executive vice-chairman, said: "In Louis van Gaal, we have secured the services of one of the outstanding managers in the game today. He has achieved many things in his career and Old Trafford provides him with a fitting stage on which to write new chapters in the Manchester United story."
In appointing Giggs as Van Gaal's assistant, United are retaining the services of a man who has been integral to that story and whose elevation from his role as player-coach is likely to prove popular.
The 40-year-old, who made 963 United appearances and won 13 Premier League titles, stepped in as interim manager for the final four games of the season following Moyes's exit. The Welshman's future is thought to have been a key part of discussions with Van Gaal.
"I am thrilled to have the chance to serve as assistant manager," said Giggs. "Louis van Gaal is a world-class coach and I know I will learn a lot about coaching from being able to observe and contribute at such close quarters. 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."
In addition to Giggs, Netherlands goalkeeping coach Frans Hoek and coach Marcel Bout, a specialist in opposition scouting, will join the backroom staff. There has been no announcement concerning the future of Moyes's former assistants Phil Neville and Chris Woods.