United had a bid of £28.85million for the 24-year-old rejected earlier on Thursday but the deal was soon finalised before Bilbao revealed Herrera had paid compensation to complete the transfer.
Herrera is the club's first major signing since Louis van Gaal - currently in charge of Holland at the World Cup - was confirmed as manager last month.
"Signing for Manchester United is a dream come true," Herrera said. "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."
Herrera has signed a four-year contract, with United holding an option to extend his stay by a further year.
Bilbao had earlier released a statement turning down a bid from United but it was thought to have been a political move aimed at appeasing the club's fans rather than a serious stumbling block.
On completion of the transfer, Bilbao said: "The Football League has certified Athletic Club that player Ander Herrera Agüera has fulfilled the requirements of the applicable regulations valid for unilateral termination of the employment contract that bound him to our club, by depositing today the amount of compensation established in the contract.
"The club thanks Ander Herrera for his dedication during the three seasons he has enjoyed at the club and wishes him the best in his future sports projects."
Herrera caught the eye at Old Trafford in March 2012 when Bilbao pulled off a shock 3-2 victory in the first leg of their last-16 Europa League tie.
The Red Devils attempted to sign the talented playmaker last summer but failed to complete the deal on deadline day with reports suggesting the transfer had been interfered with by pranksters pretending to act on the club's behalf.
Then-manager David Moyes instead bought Marouane Fellaini, who endured a disappointing debut campaign and failed to solve the team's long-term problems in central midfield.
Herrera, who scored 11 goals in 128 appearances for Bilbao, will be charged with bringing added control to the United engine-room, where he will compete for a place with the likes of Fellaini, Michael Carrick, Darren Fletcher and Tom Cleverley.
He is yet to play for Spain's senior team but has been capped at Under-20, Under-21 and Under-23 level.
"Ander is a fantastic young player, with great energy and creativity," assistant manager Ryan Giggs said.
"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."
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