There were 10 minutes left at the Millenium Stadium and Wales trailed 24-14 when they staged one of the greatest, and most thrilling, comebacks seen in international rugby.
Scotland looked as if they would claim victory in Cardiff for the first time since 2002 as John Barclay and Max Evans crossed in a breathless first 20.
But Scotland finished with two men in the sin-bin as Wales hit back with Lee Byrne and Leigh Halfpenny tries.
Stephen Jones levelled the game with a 79th-minute penalty before Williams went under the posts for a famous victory.