After losing two early wickets, including that of skipper Graeme Smith, Amla led the recovery charge with his 27th Test fifty off 78 deliveries.
Amla continued in a similar vein after lunch as he raced towards a seemingly inevitable ton, but his removal in the 47th over took the gloss off the hosts' efforts as they reached tea on 201 for five. Rahat Ali shone for the visitors with three wickets, including the key scalp of Amla.
The Proteas took an unassailable 2-0 lead into this contest after a 211-run win in Johannesburg was followed by a four-wicket triumph in Cape Town.
After winning the toss, Smith elected to bat as he began his bid to become the first ever captain to mastermind 50 Test wins.
Smith's contribution was disappointing, however, the opener managing just five runs before he was caught at second slip by Younis Khan off debutant fast bowler Ehsan Adil. The skipper's departure reduced the home side to 38 for two after Alviro Petersen (10) had earlier been trapped lbw by Rahat.
Amla and Faf du Plessis set about steadying the ship with an unbeaten 66-run partnership before the interval, with Amla doing the lion's share of the damage as he hit nine fours on his way to his 50.
Du Plessis was equally impressive as he lashed six boundaries in a 42-ball 29, but he failed to add further runs from 15 deliveries faced in the afternoon before he was caught behind by Sarfraz Ahmed off Adil.
Amla then paired with AB de Villiers and they added 79 for the fourth wicket before Amla's 13-boundary 92 was ended when he edged to Sarfraz off Rahat. Dean Elgar added just seven to the score before he fell lbw to become Rahat's third victim.
De Villiers was still at the crease at the interval on 40 not out, alongside Robin Peterson (two not out).