The Spaniard has become United's first arrival of the summer after he agreed a four-year deal with an option of a further year.
Confirmation was made on Thursday after the player paid his own buyout clause, which is said to have been in the region of £30m.
"Ander is a fantastic young player, with great energy and creativity," Giggs told the club's official website. "We believe him to be one of the brightest young prospects in Spain and I am certain he will be a big hit with the United fans.
"I'm delighted he has made the decision to come to Manchester and play his part in the club's future. I'm looking forward to seeing him in action on our summer tour to America."
Herrera is Louis van Gaal's first signing since he was appointed United manager.