Following the departure of David Moyes last month, the Welshman took over for the final four games of United's Premier League season, winning two, drawing one and losing one.
Van Gaal's appointment as a permanent replacement for Moyes was made official on Monday - the club also confirming that Giggs would become assistant to the Dutchman.
Giggs had previously hinted he could leave Old Trafford as he takes his first steps into management, but he will continue to learn his craft under the 62-year-old.
It remains unclear whether Giggs will continue as a player at Old Trafford, but the 40-year-old said he was looking forward to working under a coach of Van Gaal's experience.
"I am thrilled to have the chance to serve as assistant manager," he said.
"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."
Executive vice chairman Ed Woodward, meanwhile, felt that Giggs' long career at United would help Van Gaal to settle in once he arrives after leading Netherlands at the FIFA World Cup in Brazil.
"I'm delighted that Louis has chosen Ryan as his assistant," Woodward said.
"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."
