The left-back emerged in the Bayern team in 2009 but failed to make a lasting impact and has since been a bit-part player at the club.
The 24-year-old had been long linked to a move away from the Allianz Arena and has now completed his move to France after he underwent a medical on Monday and signed a four-year deal.
Bordeaux confirmed the deal on their official website, releasing a statement which read: "FC Girondins de Bordeaux is pleased to announce the signing of Diego Contento.
"The left back from Bayern Munich has passed his medical before signing his new contract. He is committed to the club for the next four seasons."
Coach Willy Sagnol spoke of his delight at signing the German player, saying: "Once we targeted improving the left-back position, it became clear that Contento could be interesting choice.
"He spent five years with the first team in Munich, even if he didn't play a lot of matches. We have a highly motivated player with a desire to show his true level."