After a 0-0 draw following 90 minutes in Abidjan, Williams saved spot kicks from Chancel Mbemba and Meschack Elia to give Bafana Bafana the bronze medal in the Ivory Coast.
Before Elia had his kick saved, Siyanda Xulu converted to give South Africa what proved the decisive lead in the shootout
Williams saved four shootout kicks last weekend to help South Africa get past Cape Verde after a goalless quarter-final.
DR Congo will regret many misses chances during regular time in a lively match while South Africa maintained a perfect record in third place play-offs having beaten Tunisia on penalties 24 years ago.
South Africa made three changes to the team beaten in a penalty shootout by Nigeria after an incident-packed semi-final three days ago.
Nkosinathi Sibisi, Thapelo Morena and Mihlali Mayambela came in for suspended Grant Kekana and injured duo Mothobi Mvala and Percy Tau.
DR Congo retained only two of the side that lost to hosts Ivory Coast in the other semi-final -- captain and centre-back Mbemba and midfielder Samuel Moutoussamy.
But eight of the nine who were dropped sat on the bench, leaving French coach Sebastien Desabre with multiple options should he wish to make changes.
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