c.1300, "to go, glide, move," probably from O.N. sveigja "to bend, swing, give way," from P.Gmc. *swaigijanan and related to swag (v.) and swing. The sense of "swing, wave, waver" is first recorded c.1500. The noun meaning "controlling influence"
(to be under the sway of) is 1510, from a transitive sense of the verb in Du. and other languages. The verb in this sense is recorded in Eng. from 1593.