City triumphed in the Premier League clash thanks to second-half goals from Gael Clichy and Sergio Aguero, despite Fernandinho's red card before the interval for a dangerous lunge on Johann Gudmundsson.
Sagna appeared to take issue with that and other decisions made by referee Lee Mason when he wrote on Instagram: "10 against 12.. but still fighting and winning as a team #together #mancity #youfreetothinkwhateveryouwant."
The France right-back later amended his post to remove "10 against 12" and its implication of bias towards Burnley on Mason's part, but the FA has decided he fell foul of their rules in this area.
A statement posted by the governing body read: "Bacary Sagna has been charged for misconduct contrary to FA Rule E3 (1).
"It is alleged that a comment posted on social media by the Manchester City defender constituted improper conduct in that it questioned the integrity of the match official and/or alleged and/or implied bias on the part of the match official, and/or brought the game into disrepute."
Later in the same match the 33-year-old was booked for his part in an altercation with George Boyd after Ben Mee pulled a goal back for Burnley. Clarets manager Sean Dyche said afterwards that the City man should have been sent off.
Sagna, who has made 14 appearances for Pep Guardiola's side this season, has until 6pm on Friday (local time) to respond to his charge.
Fourth-place City travels to Everton in the Premier League on Monday (AEDT) where it will once again be without Fernandinho, who serves the second match of a four-game suspension.