The 18-year-old has signed a four-year contract with the option of a further year following a transfer, which reportedly cost United in the region of £30m.
"Luke is a very talented young left-back with great potential," Giggs told the club's official website. "He has developed immensely during his time at Southampton and has all the attributes to become a top player.
"I am delighted he has joined Manchester United as I believe this is the right club for him to continue his development as a young player and eventually fulfil that promise."
Shaw returned to England two days ago following the Three Lions' failed World Cup campaign in Brazil.