The Football Association said its Independent Regulatory Commission had found Mikel guilty of the charges relating to an incident in the official's changing room.
Chelsea had complained immediately after the controversial defeat about Clattenburg using "inappropriate language" to two of their players, one of whom was Nigerian Mikel.
The FA launched an investigation into the allegations of racist language and Clattenburg was subsequently stood down from his refereeing duties before they decided that the official had no case to answer.
Mikel's ban would have been longer, the FA said in a statement, had it not been for the player's genuine belief that he had been racially abused.
"The breach of FA Rule E3 was in relation to an incident which occurred in the match official's changing room at the end of Chelsea's fixture against Manchester United on 28 October 2012," the FA statement said.
"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."
In November the FA said the evidence for the accusation against Clattenburg had come from Mikel's Brazilian team-mate Ramires.
In clearing Clattenburg, the FA said: "The details of the allegation were that following one or the other of the red cards issued during the second half of the game, Ramires heard Mark Clattenburg say to John Obi Mikel, "shut up you monkey".
John Obi Mikel did not hear the alleged comment.
"Ramires, whose first language is not English, explained that his instinctive reaction was to seek confirmation from John Obi Mikel as to what the referee had said.
"John Obi Mikel, who was being spoken to by the referee, was much closer to the referee than Ramires and did not hear what it is suggested was said to him."
FIFA referee Clattenburg has now returned to his duties and Chelsea last month issued a statement expressing regret over their handling of the allegations.
Mikel will miss Premier League matches against Sunderland and Aston Villa as well as the League Cup tie against Leeds United but the ban will not affect his participation in the Club World Cup in Japan.