They will play in their first final since 2007 when they meet England next weekend at this stadium.
In a nerve jangling semi-final dominated by high kicks and very little attacking play the teams were deadlocked at 9-all before Damian de Allende touched down for the Boks. Handre Pollard added the conversion but Wales scored a converted try in the 65th minute to level matters.
Pollard however showed nerves of steel by slotting the winning penalty four minutes from full time.
Scorers
South Africa (9) 19
Try:
Damian de Allende.
Conversion: Handre Pollard.
Penalties: Pollard (3)
Wales (6) 16
Try:
Josh Adams.
Conversion: Leigh Halfpenny.
Penalties: Dan Biggar (3).
