An initial offer of £28.85m was rejected by Athletic Bilbao, although the move was confirmed after United met the release clause in the player's contract.
Herrera - who has signed a four-year contract - becomes the first player to arrive at Old Trafford since Louis van Gaal was named as manager. "Signing for Manchester United is a dream come true," said Herrera. "I have played at Old Trafford for Athletic Club in the Europa League and it was one of the highlights of my career so far.
"I have joined United to do all I can to help the team reach the club's objectives. Having arrived on Tuesday I'm excited to now be living in Manchester and I can't wait for my first game in United's famous red shirt."