South African captain Graeme Smith ended a period of personal torment and hailed "one of the great comebacks in Test cricket history" after South Africa sealed an eight-wicket win by lunch on the third day of the first Test against Australia at Newlands on Friday.
Smith and Amla hit centuries as South Africa cruised to victory less than 24 hours after they conceded a 188-run first innings deficit in one of the most topsy-turvy Tests in history.
The South African captain praised the resilience of his team, who bowled Australia out for 47 in the second innings, leaving the home side with a target of 236.
Smith said the talk at the end of the first innings, in which South Africa were bowled out for 96, was less about motivation than specifics.
"It was about what we wanted to achieve. We never expected 21 for nine (which Australia were before a 26-run last-wicket stand), although there were a few jokes about 60 or 70 but I don't think anyone believed it. It was about a bit of direction and the bowlers knew they needed to do a job for us.
"That shows the character of the team. We didn't just back off. Obviously there was something in the wicket for us but to bowl with that intensity was a good sign."
Both Smith and man-of-the-match Vernon Philander said they took inspiration from Australia's Shane Watson, who took five for 17 to shoot South Africa out.
"Watson got the ball to talk and we learned something from that," said Smith. "He was very tight on off stump.
"There was enough movement each way to create chances. We really focused on that coming out to bowl. I thought our execution and the way the bowlers bowled was terrific."
Philander, who topped Watson's effort by taking five for 15 to record match figures of eight for 78 in his debut Test, said: "I looked at what Watson did and thought, this is my game. All I did was bowl a bit fullish on about off stump."
South Africa's victory was effectively sealed by a second wicket partnership of 195 between Smith and Amla, with Amla scoring a sparkling 112 off 134 balls before he was out when only 14 runs were needed.
Smith reached his century with a single to level the scores and then hit the winning run to finish with 101 not out off 140 balls.
It was the fourth time Smith had scored a century in a winning fourth innings run chase - more than anyone else. He also became the first man to score 1000 runs in winning chases, with an average of 85.83.
Smith said he took great personal satisfaction after what he described as "a really tough period" during which he was criticised after South Africa's failure in the World Cup, which was followed by knee surgery which kept him out of action until shortly before the Australians arrived.
He said some of the criticism had hurt. "Hopefully one or two people will know that it means a hell of a lot to me to play for South Africa. One or two things hurt a lot and were personal and maybe crossed the line.
"But it's the nature of the game. It's been a big battle to get the confidence and the self esteem back. At the end of the day I just want to talk with my skills and perform with the bat."
Australian captain Michael Clarke bemoaned his side's poor batting performance in the second innings. "Without doubt we let ourselves down with the bat. Full credit to South Africa, for how they bowled and the way they played today."
Clarke said: "Every single one of us needs to be disappointed. When you lose like that you don't just let yourselves down, you let the people who support you down.
"We need to find a way to get back up. Good teams do. That's how you learn, in my career I've learnt more from the not so good days than the good days."
The teams meet in the second and final Test in Johannesburg starting on Thursday.