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.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
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)
