The 24-year-old was expected to join the Premier League champions after the Spanish club confirmed that Herrera had paid his own buyout clause.
Despite the La Liga outfit initially rejecting a £29m bid from the former Premier League champions, the move has now been completed after Herrera passed a medical and agreed personal terms earlier this week.
United announced on their official website that the midfielder has signed a four-year contract, with an option of a further year.
"Signing for Manchester United is a dream come true," Herrera told the club's official website. "I have played at Old Trafford for Athletic Club in the UEFA 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.
"I will never forget my time with Athletic Club and I'd like to thank the fans and staff for all their support and, above all, wish all my fantastic former teammates all the best for the future."
Herrera is Louis van Gaal's first signing as United manager.