Chasing a modest 149 to win in testing batting conditions, Amla took his side home shortly before tea on 63 not out to rediscover his form with a second half-century in his last 14 innings.
He put on 119 for the second wicket with the gritty opener Dean Elgar (50) to lay the platform for the win, though both survived early scares as Pakistan's seamers created immense pressure on a wicket that showed both sideways movement and variable bounce.
The second test begins in Cape Town on Jan. 3 with the series to conclude in Johannesburg on Jan. 11.
"It was a good test match and a good one to win in terms of the series," South Africa captain Faf du Plessis said at the post-match presentation.
"We needed a bit of luck on a wicket like this one. It was challenging, the first two hours was proper test cricket.
"The Pakistan bowlers were bowling really well and we felt like we were going nowhere.
"But I’m really chuffed for Hashim, he looked like he was moving nicely and that is always a good sign."
Starting their second innings chase under gloomy skies after overnight rain, South Africa opener Aiden Markram was trapped leg before wicket by Hasan Ali (1-39) for a duck before the home side had a run on the board.
Pakistan bowled with skill and accuracy, and beat the bat on numerous occasions as Amla was dropped on eight at third slip by Fakhar Zaman.
Pakistan's irritation grew when moments later they thought they had Elgar caught at first slip when he was on four.
Seamer Shaheen Afridi (1-53) induced the edge that looked to have been gobbled up by Azhar Ali. Elgar began walking towards the dressing room, but was called back when the on-field umpires conferred over whether the catch had been taken.
It was sent to third umpire Joel Wilson with a soft signal of 'out', which called for conclusive evidence for the decision to be overturned.
With only the two-dimensional images available, the West Indian decided there was enough doubt to change the decision.
It was a potentially a game-changing moment in the innings with South Africa on 16-1 at the time and struggling.
GOOD FORTUNE
Elgar and Amla capitalised on that good fortune and both would go on to make half-centuries in a stand that would last 40 overs.
Elgar (50) then handed a first test wicket to Pakistan part-timer Shan Masood when he had a swipe at a loose delivery and was caught by wicketkeeper Sarfraz Ahmed.
Theunis de Bruyn (10) was stumped off Yasir Shah (1-20) trying to finish the game with quick runs and Du Plessis recorded a pair of ducks in the match when he attempted a hook off Afridi and was caught at fine leg.
Pakistan skipper Ahmed focussed on his side's poor batting, where they failed to pass 200 in either innings and lost their last nine wickets for 90 runs after tea on day two, as the major contributor to their defeat.
"We are very disappointed in the batting, we had an opportunity to post a (bigger) target and we lost too many wickets in that session after tea," he said.
"We have good quality fast bowlers, but we have to put scores on the board. We will have to go away and work hard in the nets."
(Reporting By Nick Said; editing by Sudipto Ganguly and Christian Radnedge)
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