Middlesex had been searching for a replacement after Paul Downton was confirmed as the England and Wales Cricket Board's new managing director, and Strauss, who played domestic cricket with the club for 14 years, has been given the responsibility.
"We are absolutely thrilled that Andrew has accepted our invitation to join the board," Middlesex chairman Ian Lovett told the club's official website.
"He is a Middlesex man through and through, with an outstanding reputation within the game.
"Finding the right man to replace Paul Downton was always going to prove a challenge. However, in Andrew Strauss, we've found the perfect solution."
Strauss made his first-class debut for Middlesex in 1998, before going on to represent England 100 times in Test cricket.