Williams said: "I would not say we felt humiliated but I am very angry at the performance. We just did not play like we did in the Six Nations.
"We are a better side than that and need to show it next week. The display was very disappointing and atrocious.
"We have trained really well over the past few weeks and we just let ourselves down. We just did not play like how we can.
"We knew South Africa would be very physical. But we have worked really hard in training and the gym and felt we would be able to compete.
"We just could not hold on to the ball and gave them too many turnovers."
Wales contributed to their downfall with a high error count that handed the Springboks a steady stream of points on a plate.
It was a bitterly disappointing performance that added up to coach Warren Gatland's first defeat since taking charge in January.
Assistant coach Rob Howley has refused to excuse the Bloemfontein no-show but believes Wales will benefit from the defeat.
He said: "We've tried to take as many 'positives' as we can from that - but it's not easy. The speed of the ball in international rugby is really important.
"They dominated physically and dominated the tackle area very well. They had quick ball, and the runners to use that ball efficiently.
"South Africa had quick ball and we had slow ball. We need to shore up in that area - you don't win Test rugby when you turn the ball over 18 times.
"We also gave away seven penalties inside the opening quarter. These are the areas of the game we must improve on - and we know that.
"That will happen, and we'll be better for this experience next weekend.
"Test rugby is a sharp learning curve, and we'll all learn - players and coaches - from this."
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