Television replays showed Francis got a touch of the ball before bringing down Leicester's Jamie Vardy, resulting in a penalty which Bournemouth goalkeeper Artur Boruc saved.
"The defender, who was dismissed for denying an obvious goal scoring opportunity against Leicester, will serve his one-match suspension with immediate effect," the FA said in a statement.
Currently four points above the relegation zone after 20 matches, Bournemouth will make a trip to Championship side Birmingham City on Saturday as the Cherries seek to progress to the next round of the FA Cup for the third season in a row.
Francis will be eligible to feature in next week's Premier League fixture against an in-form West Ham United at the Vitality Stadium.
Manager Eddie Howe will have right-back Adam Smith available for the FA Cup clash at St. Andrews.
(Reporting By Arnab Paul in Bengaluru; editing by Amlan Chakraborty)
