Everton lost 2-1 at Goodison Park in controversial fashion and Moyes was unhappy at the performance of referee Mark Clattenburg after he awarded the visitors two penalties.
The match official also came under fire after the failed to award Everton what Moyes believed was a clear penalty in stoppage time.
"Having considered all of the evidence and submissions, the commission concluded that the comments would reasonably be interpreted as calling into question the referee's integrity and implying that he was motivated by bias," read an FA statement.
"The commission recognised the charged atmosphere of the game and the frustration that Mr Moyes was feeling. It also took into account Mr Moyes' exemplary disciplinary record.
"However, the commission also stated that it is very important for managers to exercise great care in making comments about referees as they set an example for others."