The Bible describes an annual ritual in which one goat was selected for sacrifice and another had the sins of the people symbolically laid upon it before being set free, taking those sins away into the wilderness.
This goat became known as the scapegoat. Scapegoat has a similar meaning today, although it's typically applied to people, not actual goats. Anybody who is blamed for a bad situation or outcome — whether they deserve it or not — is known as a scapegoat.
The following examples show scapegoats in the wild. In the situations described below, one individual takes all the blame, even if there's plenty of guilt to go around.
There was always one unfortunate whom the teacher singled out and used for a scapegoat. (A Tree Grows in Brooklyn)
The lawyer representing the referee said police have no real evidence to justify her client's detention and that investigators are trying to make him a scapegoat. (Seattle Times)
Escape goat is an "eggcorn," a word or phrase that is a mishearing of another word or expression. Scapegoat is one word and has a specific meaning related to its Biblical origins. Escape goat seems to refer to a farm animal that's good at sneaking out of its pen, not someone who is blamed for the mistakes of others.
The next two examples show how silly escape goat looks and sounds when you know the correct word.
Antonio Pierce, on ESPN, was talking about how the Washington Redskins seemed to be blaming their quarterback for a lot of their troubles. "I think they're using Donovan McNabb as an escape goat," he said. (Columbia Journalism Review)
I once heard someone talk about his office being made the escape goat for problems. (Bad Language)
Ironically, this ancient term is in fact likely based on a misreading of the original Hebrew, but that's another story. After 2,500 years, it's much too late to find a scapegoat for this Biblical bungle.