There are a few different ways to approach this.
First, sacrifice is the normative means of atoning for sins. In the absence of this primary method, other means can be used. Let me offer an analogy. After a person becomes a Christian, it is normal that the person will be baptized. However, let's say the person is walking to church for his baptism, and it hit by a truck and killed. Is he saved? Yes, because while baptism is normative, it is not absolutely necessary.
Second is a variation of the first. It recognizes that sacrifice is normative, but considers our prayers asking for forgiveness to be a 'substitute' for the sacrifice--a "sacrifice of the lips".
Remember that even under the sacrificial system, a person had to repent and ask forgiveness. A sacrifice without repentence was worthless. The sacrifice is a symbolic act of offering something of value to God as a sign of our contrition or gratitude.