Religiously, sacrifice usually means forcing the gods to do what you want them to do by subduing them through a pleasing offering. The irony here is that the gods in many civilization (particularly ancient Sumer) were thought to get sleepy from eating the oblations. Hence, the logical extension of the pacification of the gods is to make them literally sleep- or perhaps even die. To my mind this seems to express the essence of Original sin. Humans wish to be God, so they try to pacify the God. The ultimate act of pacifying God would be to kill him- hence the crucifixion!
Christianity is different for two reasons. First, Christianity says that the world is not rooted in violence, but in God's peaceful act of giving (creation ex nihilo). Secondly, sacrifice does happen, but at the end of history and not that the beginning. In other words, in a sense the pagan myths are right. Human community after the Fall does have to be rooted in some group of orderly people forcing disorderly people to be orderly. Moreover, evil really does need to be punished. Hence the curse and condemnation of the law. In the punishment of sin in the cross, the point is to restore the primal peace between God and humans. Sacrifice is an interim measure to deal with evil, it is not the very basis of reality. Of course the sacrifice of cross does found a community, but one based on the end of sacrifice and therefore the end of the project of human self-justification.