Kone scored the opening goal of the fourth-round contest after less than two minutes, but during the celebrations some Everton players complained of racial abuse aimed at the striker.
The police have already confirmed that they are looking into the incident, and the FA have now revealed that they will also launch their own investigation into the events at Brunton Park.
"We have received the referee's report regarding the alleged incident during today's game between Carlisle United and Everton," read a statement from the FA.
"We have contacted both clubs this evening and will be conducting an immediate and thorough investigation.
"The FA strongly condemns any discriminatory behaviour and will continue to work with partners across the game to combat it whenever it arises."