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Rusty Theron and Wayne Parnell shared an unbroken 64-run eighth-wicket stand as the Proteas, needing 148 to win, fought back to secure a three-wicket victory with five balls to spare at the Wanderers.
"At 84/7 a win looked very far away for us but the guys were pretty calm on the bench," Amla said.
"It helped that we didn't get too flustered."
Amla praised Theron, the Man-of-the-Match, and all-rounder Parnell as the hosts levelled the two-match series at 1-1.
"It was sensible batting at the end, mixed with big hitting, which won us the match," he said.
"It's a tribute to them and to what we are trying to develop in the team."
Theron smashed 31 runs off 16 balls, finishing off the match with a big six, while Parnell contributed 29 from 11 deliveries.
Theron also took 2/28 in Australia's innings, and although Parnell did not have a good day with the ball, conceding 11 runs an over, Amla said he was developing as a genuine all-rounder.
"We always knew Parnell could bat and he'll bounce back as a bowler," the skipper said.
"He's still got a lot of improvement to do but we know he can do it."
While Dale Steyn and Jacques Kallis were rested for the T20 series, and AB de Villiers was recovering from surgery, it gave Amla and new coach Gary Kirsten a perfect opportunity to groom a few of the younger members of the squad.
"Of course the senior players will be missed as they're such class players," Amla said.
"But they were rested to give others an opportunity. We already know what fantastic players [the seniors] are but we need to get an idea of how the youngsters play."
He admitted, however, it was difficult to gauge the less experienced players in this format of the game.
"T20 is difficult to form a judgement but our youngsters have a great attitude and a massive desire to learn," he said.
"That's a combination that can take you really far and we're lucky they're in our team and doing well.
"Everyone is behind them and the team is in a good space to allow them to express themselves."
Steyn and Kallis joined the squad on Monday ahead of the three-match ODI series which starts in Centurion on Wednesday.
Veteran wicketkeeper Mark Boucher, spin bowler Imran Tahir and all-rounder Faf du Plessis also linked up with the squad.
Amla said the T20 win had lifted the team and would stand them in good stead ahead of the ODI series.
"It's always important to get a victory," he said.
"The teams will change for the one-day series, but the spirit of the team is lifted."
