The fundamental sacrifice is that of the will.

The fundamental sacrifice in Judaism is that of the will. Since freedom of the will is the highest gift of God to man, the way that we acknowledge that it is a gift is periodically to give some of it back. …

What matters in sacrifice is the act of renunciation. We give up something of ourselves offering it to God in recognition of the gifts He has given us. It is remarkable how readily the sages found substitutes for [animal] sacrifices, most notably in the form of prayer, … charity, and hospitality.

Rabbi Jonathan Sacks, Covenant & Conversation: Leviticus

Of course, this applies to Christianity as well.