England struggled to make headway against a fierce Springbok defence, although it took the video referee to rule out Mark Cueto's 'try' early in the second half for a foot in touch.
"We had a lot of ground to catch up in this tournament and the guys all took the responsibility and I was proud of them all tonight," Wilkinson said of his team-mates, who lost 36-0 to the same opponents in the pool stage.
"I have given everything I had to this effort. I have never given anything less than everything and it has been a hell of a journey with these guys.
"We have played pressure rugby for all of this tournament and even today there were only a few points in it."
Wilkinson spoke to Prime Minister Gordon Brown on the rostrum after the game.
He added: "The Prime Minister said: 'Well done. All the boys back home are proud of you. Unfortunately it just wasn't your day today.'"
Cueto insisted there was no question he should have been awarded a try after finishing off a brilliant England attack, sparked by Mathew Tait's midfield break.
He grounded the ball in the left corner but, after numerous replays, video referee Stuart Dickinson ruled he had brushed the touchline as Danie Rossouw attempted a last-ditch tackle.
It came at a crucial moment, straight after half-time with England 9-3 down.
Cueto said: "I watched it on the replay and I thought it was a 100% try.
"From the front the ball was grounded well. From the back you could see I wasn't in touch.
"The foot they thought was in touch lifted up before it went over the line so for me it was a 100% try.
"It was a crucial point. We were six points behind. With the try we would have been within one with Wilko to take the kick.
"It was a tight game. Neither side looked like scoring, so it was a key moment."