The 23-year-old, who can play either as a defensive midfielder or in the centre of defence, has signed a five-year contract with the Bundesliga giants.
Bilbao, who had earlier claimed that Martinez had left to discuss terms and undertake a medical in Munich without their permission, will receive a £32 million fee from Bayern which matches the buyout clause in the player's contract.
Bayern CEO Karl-Heinz Rummenigge revealed that the player helped push the move through by accepting a lower salary than he might have otherwise expected.
"Of course this is a large sum," Rummenigge said via the club's official website, "but the player participates indirectly in it. He has given up large sums in his salary."
The new signing has been assigned the No.8 shirt, and could make his debut in Sunday when Bayern host Stuttgart at the Allianz Arena.
Martinez, a youth product of Osasuna, leaves Athletic Bilbao having made 201 La Liga appearances since making his senior debut in 2006.
Last season was his most successful at the club as Marcelo Bielsa's side reached the finals of both the Copa del Rey and the Europa League.
While Athletic lost both those finals, Martinez did enjoy success this summer when he was part of the Spain squad that won Euro 2012, enabling him to add a European Championship winner's medal to the one he earned at the World Cup in 2010.