From a letter by Secretary of State William H. Seward to the President (source) Any emphasis is mine:

Western Virginia is organized unquestionably with all the constitutional elements and faculties of a State, and with a republican form of government. It, therefore, has a title to be a candidate for admission into the Federal Union. Congress has power to admit new States, but is a power restricted within certain limitations. One of these limitations is that no new State shall be formed or erected within the jurisdiction of any other State without the consent of that State as well as the consent of the new State and the consent of Congress. It is an undisputed fact that the new State of Western Virginia has been both formed and erected within the jurisdiction of the State of Virginia. Has the consent of the State of Virginia to the formation and erection of the State of West Virginia been given, or has it not been given? Upon this point the constitutionality of the Act of Congress now before me turns. The constituted and regular authorities of a State called the State of Virginia sitting at Wheeling, within the jurisdiction of that State, claiming to be the State of Virginia, and acting as such, have in a due and regular manner declared and given the consent of the State of Virginia to the formation and erection of the State of West Virginia within the jurisdiction of the State of Virginia. Thus far the case seems simple and clear. But it is just at this point that a complication begins. If we would unfold it successfully we must first state the existing facts in regard to the constitutional position of the State of Virginia, as well as those which belong to the formation and erection of the new State of West Virginia.

The TL:DR is that because the government sitting at Wheeling VA was recognized by the US as the legitimate government of the State of Virginia and that government had given its consent, West VA was therefore considered to have VA's blessing to leave.

For further reading, I'd suggest looking up Virginia v West Virginia, which settled this question in the Supreme Court. link to the wikipedia page