Essien was shown a red card for pushing Kenny Miller in the face during stoppage time at Derby and he will now miss games against West Ham, Sunderland as well as the clash with Premier League leaders Arsenal.
The dismissal was the only blot on Chelsea's fourth consecutive away win, which was secured with goals from Salomon Kalou and Shaun Wright-Phillips.
"We are not happy with the decision with Michael Essien, but I don't want to go shouting my mouth off because I will get into trouble," Cole said.
"From what I saw, it did not look like he really got him.
"He does that every day in training. It is just the way he is. He does not mean to hurt anyone. He just plays hard. We all know how strong he is and he is not a nasty player.
"It is a big blow to lose him for a suspension now, because we have got a lot of tough games coming up and it is going to be tough without him, but Steve Sidwell did brilliantly and it is great to have him to come in."