It was a thrilling finale to a superb encounter, Davies setting up a late try for Ashley Evans and then adding the extra two points to reverse what had been a 17-11 deficit.
Wales will now face England in Sunday's final but before focusing on taking home the trophy, Wilson says his side must enjoy that winning feeling.
"Tonight the squad can relax and savour what they've done but we're keeping our feet on the ground for some hard work ahead," said Wilson in the immediate aftermath of the win at the Stade de la Rabine.
"To beat South Africa is a great feeling but we'll be sticking to our usual recovery and analysis though this time we know we're preparing for a World Championship final.
"Parts of our game were excellent and although we have a few things to work on, we know what we have to do now in the final."
The Wales head coach had seen his side more than match a powerful South African pack in the first half although they did go into the break 7-6 behind. Davies had kicked two penalties, South Africa responding with a converted try from lock Irne Herbst.
"I don't know how many tackles we put in against the South Africans but it was a huge defensive effort and I congratulate our boys for that," said Wilson.
"We've had to change our game plan in a few of our games so far. But I'm pleased with the way the boys have responded and stuck to the patterns we have given them."
In a thrilling second half, captain Ellis Jenkins put Wales back into the lead before Kwagga Smith replied for the Springboks.
But inspired by Jordan Williams, who produced a mesmerising performance from full back, Wales secured the win at the death courtesy of Evans' converted score.
"It's difficult to single anyone out but it's fair to say Jordan has played exceptionally," said Wilson on the Scarlets No. 15.
Jenkins, who once again led his side superbly throughout, added: "It's an unbelievable feeling and all the boys deserve huge credit for putting in such fantastic performances.
"To be six points down and then to come back on virtually the last kick of the game was a very special way to get the result."
The Wales captain, who guided Wales to second place in this year's Six Nations, was understandably delighted at the final whistle.
And he reserved special praise for the tight five in front of him, the Welsh forwards playing a key role in today's win.
"Our scrum against Argentina and South Africa has been fantastic and I'm proud of all of our front five for what they did tonight," said the Cardiff Blues man.
"That platform allowed our half backs to keep to our game plan and put together some good attacks.
"We know the final is going to be huge but it's a challenge we are really looking forward to."