One centuries-old meaning of the noun sway is power or controlling influence. This is the sense intended in hold sway and similar phrases. To hold sway (usually over something) is to have and exert power and influence. Often the phrase implies that the person or thing holding sway prevails over other competing influences.
Whatever the merits of competing economic models, there is no question that in the United States, cutthroat capitalism holds sway. [New York Times]
The “modern infrastructure ideal” that held sway in the metropolis usually did not survive the trip to the colonies. [Megacities]
[I]t gives the Shia movement an easier connection between Lebanon’s eastern Bekaa Valley, where Hezbollah holds sway, and the major Syrian cities. [BBC]
The police also were saying that nobody had stepped forward to assist them in their investigation, the street edict against snitching apparently holding sway even when the victim was a baby. [Daily Beast]