Former Ajax Amsterdam, Barcelona and Bayern Munich coach Van Gaal, who will lead Netherlands at the World Cup next month, has agreed a three-year contract, United said. (Agencies)
Giggs, the club's record appearance maker, took charge of United's final four Premier League of the season following the sacking of David Moyes.
Van Gaal has won domestic titles and domestic cups in three countries, as well as the Champions League with Ajax and the UEFA Cup.
"In Louis van Gaal, we have secured the services of one of the outstanding managers in the game today,” United executive vice-chairman Ed Woodward said in the statement.
"He has achieved many things in his career to date and Old Trafford provides him with a fitting stage on which to write new chapters in the Manchester United story,” he added.
"Everyone is very excited about this new phase in the club's history. His track record of success in winning leagues and cups across Europe throughout his career makes him the perfect choice for us.”
"I'm delighted that Louis has chosen Ryan as his assistant. Ryan's association with the club spans over two decades and his knowledge and stature will be of great use to Louis.”
"In addition, this is a fantastic opportunity for Ryan to learn his trade alongside a world class manager whose attacking instincts and belief in youth are tailor-made for Manchester United,” he further said.
Van Gaal will also bring Netherlands' goalkeeping coach Frans Hoek and scouting specialist Marcel Bout to the club as assistant coaches.
"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.
"This club has big ambitions; I too have big ambitions. Together I'm sure we will make history,” he added.
United, English champions on 20 occasions, finished seventh in the Premier League and will miss out on European football next season.
Moyes, who replaced the retired Alex Ferguson, was sacked last month after 10 troubled months in the Old Trafford hot seat.
Former Ajax Amsterdam, Barcelona and Bayern Munich coach Van Gaal, who will lead Netherlands at the World Cup next month, has agreed a three-year contract, United said.