South Africa went 20-17 ahead with six minutes left but some sloppy defending allowed Wales to grab the late try and snatch victory.
Wet conditions could be blamed for an appalling start to the test as the opening half hour saw a string of handling errors and scrum infringements, before Wales burst into the lead with two tries in the space of three minutes for Amos and Williams.
The first came as Wales captain Ellis Jenkins sprung onto the ball as it emerged from a ruck to set up a counter-attack with a quick series of passes that saw fullback Amos go over.
The 31st minute try was followed by a score for Williams as the Springboks defence was easily breached, the Welsh scrumhalf slipping two tackles before crossing the line.
The Welsh 11-point half-time lead was cut just after the break with a fortuitous intercept try for Ismaiel, one of seven new caps in the South African starting line-up.
Gareth Anscombe’s penalty put Wales 17-10 ahead, after which he missed another easier chance and the Wales backline dropped the ball with a massive overlap out wide when they looked sure to score.
South Africa pulled level on the hour mark as left wing Mapimbi went over on his debut and Elton Jantjies converted.
South Africa then went ahead with time ticking away as flyhalf Robert du Preez, who had come on only seconds earlier, booted over a penalty.
However, his attempted clearance behind his own line was charged down and Elias pounced on the ball to put Wales back ahead 22-20.
The match marked a losing start for new Springboks coach Rassie Erasmus, who left behind his first choice players in South Africa ahead of a three-test home series against England, which starts in Johannesburg next Saturday
South Africa went into the match with 28 wins in 33 tests against Wales but have now lost three of their last four against them. Wales head next to Argentina for two tests.
(Reporting by Mark Gleeson in Cape Town, editing by Peter Rutherford)
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