But each time Pollard denied them the chance for what could have been be a crucial losing bonus point, landing a drop goal the first time and a 50-metre penalty the second time.
It was the first defeat in the tournament for the Scots and for South Africa it made their pathway from Pool B to the play-offs more clear.
The opening loss to Japan has been put behind them as they improved to two wins from three games, the same as Scotland and Japan who posted their second win when they ended Samoa's long-shot hopes earlier on Saturday.
With South Africa and Japan still to play the United States, and Scotland's final game against Samoa, the pool could well be decided on bonus points.
South Africa, who missed an attacking bonus point against Scotland, have three, Scotland two and Japan nil.
Halfback Fourie du Preez, stepping up to captain South Africa after the inspirational Jean de Villiers suffered a career-ending broken jaw against Samoa, directed play with surety.
He used South Africa's pack, led by man-of-the-match Lodewyk de Jager, to out-muscle the Scots, dominating scrums and mauls, while raking kicks from five-eighth Pollard were used to turn Scotland wings Tommy Seymour and Tim Visser.
Wearing their white away strip, the Springboks announced their forward dominance from the start as they set up camp deep in Scotland territory.
Even after being driven back to their own 22 by a couple of quick penalties, a tighthead allowed them to ram their way back up field with a try awarded to Schalk Burger after a mass drive over the line.
The signs were ominous for Scotland that they were well outmuscled up front.
A forward drive that marched the Scots back nearly 15 metres eventually produced a penalty which Pollard duly slotted from in front of the posts.
Facing an early 10-0 deficit, Scotland tried to up the pace of the game to upset the Springboks rhythm but for every metre they gained by running the ball they lost many more as South targeted the Scots wings with deep kicks.
South Africa led 20-3 at the break.
Scotland struck back at the start of the second half with a quick 10 points from another Greig Laidlaw penalty who also converted Seymour's try.
The wing was on hand to finish after Duncan Weir, starting at five-eighth in place of the injured Finn Russel, pulled off a well-time intercept when Scotland were desperately defending five metres from their line and ran 80m downfield before unloading for the wing to score
With the deficit at seven points, Pollards drop goal stretched it to 10.
When Laidlaw landed his third penalty, Pollard came back with two more.
With time running out Brian Habana crossed for South Africa's third try, but the vital fourth for a bonus point eluded them.
- AFP
South Africa 34 (Schalk Burger, JP Pietersen, Bryan Habana tries; Handre Pollard 2 conversions, 4 penalties, dropped goal), Scotland 16 (Tommy Seymour try; Greig Laidlaw conversion, 2 penalties, Duncan Weir penalty). HT: 20-3.
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