Williams was sent off for two bookable offences
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A scrambled goal on the stroke of half time saw lowly Bristol City to victory.
High-flying Swansea failed to react to an Alex Russell free-kick, and with keeper Willie Gueret rooted to the line, defender Louis Carey shot home.
Swansea wide man Tom Williams was booked for unsporting behaviour in a bad-tempered first-half by card-happy referee Clive Penton.
An awkward 83rd-minute challenge on the outstanding Dave Cotterill then led to a red as the home side eased home.
Swansea face a club fine after having seven players booked in the game.
Bristol City: Basso, Orr, Heywood, Carey, Woodman, Cotterill, Noble (Skuse 33), Russell, Murray, Brooker, Savage (Fontaine 90).
Subs Not Used: Phillips, Stewart, Brown.
Booked: Orr, Brooker.
Goals: Carey 45.
Swansea: Gueret, Watt (Britton 61), Iriekpen, Monk, Austin, Forbes (Knight 79), Martinez (Way 61), Robinson, Williams, Trundle, Fallon.
Subs Not Used: Murphy, Akinfenwa.
Sent Off: Williams (83).
Booked: Monk, Watt, Martinez, Robinson, Williams, Iriekpen, Way.
Att: 12,859
Ref: C Penton (Sussex)