Synonyms for Sway:

n.

bounce, flutter. toing and froing, wiggle, roll, quiver. woo, convert, sweeten up, target, reason with, get in with. authority (noun)
commission, faculty, enfranchisement, credential, prerogative, kingship, primacy, purview, entitlement, stature, precedence, rank, powerfulness, potency, superiority, steam, office, strength, mightiness, mandate, right, sovereignty, jurisdiction, privilege, prestige, charge, cogency, domination, punch, license, title, puissance, authority, Prepotency, might, force, mastery, birthright, empowerment, seniority, clout.
energy (noun)
puissance.
fluctuation (noun)
undulation, oscillation, wavering, swinging, wave, vibration.
influence (noun)
pull, predominance, motivation, importance.
power (noun)
clout.
strong influence (noun)
control, government, jurisdiction, reign, command, dominion, might, authority, mastery, clout, rule, sovereignty, power, predominance.
sway (noun)
nod, carry, shake, swing, rock, shift, persuade, careen, wobble, tilt.

v.

coax (verb)
charm, cajole, seduce, urge, implore.
command (verb)
direct, coordinate, govern, manage, dominate.
hang (verb)
droop, drape, nod, swing, loll, sag, hang, dangle, Draggle, trail.
influence (verb)
command, empower, influence, predominate, bias, control, master, affect, lead, authorize, prejudice, pressure, motivate, power.
influence, affect (verb)
direct, carry, induce, persuade, govern, manage, dominate, bias, win over, sell.
motion (verb)
swing.
move back and forth (verb)
fluctuate, pulsate, swing, vibrate, stagger, wave, roll, wobble, rock, undulate, careen, oscillate.
oscillate (verb)
librate, radiate, alternate, throb, beat, teeter-totter, rock, undulate, wag, vibrate, pulsate, oscillate, tremble, vacillate, totter, wobble, stagger, seesaw, bob, tick, fluctuate, pulse.
persuade (verb)
allure, indoctrinate, coax, persuade, induce, sell, win over, enlist, tempt, entice, convince.

Other synonyms:

vibration, wiggle, dominion. oscillation, toing and froing, bounce. convert, roll. target. fluctuation
undulation.
oscillate
teeter-totter, wiggle, librate.
Other relevant words:
flutter, swinging, oscillation, quiver, carry, toing and froing, wavering, shake, roll, vibration, wiggle, undulation, wave, convert, jurisdiction, careen, shift, dominion, tilt.

Usage examples for sway

  1. And there there was no prison, power of arms, nor palace, Where prince or judge held sway for none was needed there; Long ages since the very names of fraud and malice Had vanished from men's tongues, and died from all men's care. – Alcyone by Archibald Lampman
  2. It is all so bare and prosaic; the flat stones lie upon the graves as if there was a fear lest, if not held down in such brutal fashion, the wretched dead would rise and return to a world where there is no longer any place for them, and where interests hold sway in which they have no part. – 'O Thou, My Austria!' by Ossip Schubin