The incident in the 95th minute of the match was missed by referee Lee Probert, but was caught by the FA after a review of video evidence.
Gomes will miss Everton's league games against Manchester United, Crystal Palace and Burnley.
The 25-year-old will be eligible to return for the Merseyside club against Tottenham Hotspur on the final day of the league campaign on May 12.
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.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
