The midfielder entered the referee's dressing room after the 28 October clash at Stamford Bridge to confront match official Mark Clattenburg.
Mikel had been informed by team-mates that Clattenburg had used inappropriate language of a racist nature towards him during the match, allegations which later proved to be false.
The Football Association acknowledged the circumstances that led to his actions and said the levels of the ban and fine had been imposed accordingly.
A statement on the FA website read: "The Regulatory Commission's independent chairman Christopher Quinlan QC emphasised that the Independent Regulatory Commission accepted, as did The FA, that at the time he threatened the referee the player genuinely believed that the referee had racially abused him.
"But for that factor the suspension would have been significantly longer."
Mikel accepted the FA's charge of "using threatening and/or abusive and/or insulting words and/or behaviour in the official's changing room" and will now miss the Premier League games against Sunderland and Southampton as well as the Capital One cup quarter-final against Leeds.
The referees trade union, Prospect, said they were dismayed at the "leniency" of a three-game ban.
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