Kompany was shown a red card for protesting against a penalty decision when Burnley's Lorenz Assignon (23) received a second yellow card for a foul, reducing the visitors to 10 men in the match at Stamford Bridge on March 30th.
The Belgian will serve a touchline ban for the home game against Brighton & Hove Albion on Saturday, with the second match suspended until December 31st.
"The manager admitted that his language and/or behaviour in the 40th minute of the fixture was improper and/or abusive and/or insulting towards a match official and/or questioned the integrity of a match official," the FA said in a statement.
Kompany was also fined 10,000 pounds.
Burnley are second-bottom, six points from the safety zone with six games left this season.
