The normally placid Villa striker also said that he had NOT been provoked by racist remarks from Savage.
Dublin told Sky Sports News: "What was said is irrelevant. My actions were wrong. It was the wrong reaction.
"I am ashamed of what I have done. I've let the players, the manager and staff down - and most importantly I have let myself down.
"I called a meeting of my team-mates this morning and said sorry to them. I wanted to say sorry to their faces.
"I have never done anything like that in my career before. Where it came from, I just don't know. I just don't know why I reacted like that.
"Possibly it was partly frustration but I have got no explanation for my actions. But it won't happen again."
Dublin was captured on camera muttering the words "he's a cheat" towards the television cameras as he came off the pitch and when questioned on that subject, said: "I didn't really know what I was saying at that stage. I was so angry with myself just for letting myself down.
"But I can clear up the fact there was no racial element at all to this incident.
"Words are exchanged in football in the heat of the moment. But I should not have reacted that way."