Austin admitted making the two-fingered action as he was substituted in the second half of Southampton's 3-1 defeat against City at St Mary's in December.
"The striker admitted his gesture in or around the 68th minute of the game against Manchester City on 30 December 2018 was abusive and/or insulting," an FA statement read.
Austin's absence will be a blow to struggling Southampton as they battle to avoid relegation.
In 2013, Jack Wilshere, then playing for Arsenal, showed the middle finger to Manchester City fans in a 4-2 defeat at the Etihad Stadium, resulting in a two-game ban.