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Williams had already come up with one vital stop at the end of normal time and the match went to penalties as it finished 0-0 after 120 minutes in Yamoussoukro.
He then produced a remarkable performance to keep out penalties by Bebe, Willy Semedo, Laros Duarte and Patrick Andrade as South Africa - who themselves failed to convert two spot-kicks - progressed to a last-four showdown with Nigeria on Wednesday.
"We were not lucky because we had a very good 'keeper today. If you can stop four penalties it is not luck anymore," said South Africa coach Hugo Broos.
Meanwhile, Cape Verde go home at the end of a campaign in which they won a Cup of Nations knockout tie for the first time, although they will regret their poor performance in the shoot-out after having the best chances during the match itself.
