to hold the rightful authority or right of control over someone or something; when turned around, to be mastered by it
Exousiazo is the verb formed from the noun ἐξουσία, which means authority or the right to act. If the noun names the authority a person holds, this verb is that authority in action: to hold the right of control over someone or something. In the everyday Greek of contracts and legal documents it was a technical term for having the right to dispose of property, whether a plot of land, a tomb, or a bride's dowry, and that background of a rightful, ownership-like claim stays close to the surface wherever the word appears.
The New Testament uses it in three settings, and the context colors each one. When Jesus describes the rulers of the nations "who exercise authority" over their people (Luke 22:25), the word carries the weight of worldly power, the very kind of ruling over others that he tells his followers to reject. Paul then turns the same verb into a legal claim about the body in 1 Corinthians 7:4: in marriage, he says, a husband and a wife each hold a rightful authority over the other's body, and he states it in perfectly matching terms for both, a mutual claim rather than a one-way command.
The word can also be turned around, so that the person is the one being controlled rather than the one in control. In 1 Corinthians 6:12 Paul writes that he will not be mastered by anything, playing on the sound of a related verb meaning "it is permitted": everything may be permitted to me, he says, but I will not let anything take permission over me. Where a harsher, overbearing note is wanted, Greek reaches instead for the compound κατεξουσιάζω, to throw one's weight down over others.