Amla reached his 20th ODI hundred in his 108th innings, faster than any other batsman, beating India???s Virat Kohli who brought up the landmark in 133 innings.
He went on to score 159 from 128 deliveries, with 16 fours and four sixes as he and Faf Du Plessis ??? who hit a run-a-ball 109 ??? made the Ireland bowlers toil at the Manuka Oval.
In reply, Ireland were skittled out for 210 as South Africa???s fast bowling trio of Dale Steyn (2-39), Kyle Abbott (4-21) and Morne Morkel (3-34) all fired.
Having won the toss and elected to bat, South Africa lost an early wicket as Quinton de Kock departed for another low score ??? eventually given out on review after edging John Mooney behind to Gary Wilson for one.
That brought the Proteas captain Du Plessis to the crease and he and Amla were at their imperious best in a 247-run partnership for the second wicket, although Amla was dropped on 10 as he flicked Kevin O???Brien???s first ball to short mid-wicket where Ed Joyce shelled the reasonably simple chance.
Ireland were made to pay dearly for the lapse as seamer Max Sorensen conceded 24 runs in one over, with Amla smashing him for two sixes and a four and Du Plessis also hitting a maximum. And having passed his hundred, Amla also took 26 runs off one Mooney over in the batting powerplay.
Du Plessis fell a few balls later, bowled by O???Brien in the 39th over with the score on 259 ??? but the perfect platform for the incoming AB de Villiers had been set.
He couldn???t repeat his own 150-run heroics from the win over the West Indies ?????managing only 24, although that did come in just nine balls, with a four and two sixes..
He fell two balls after Amla, in the same Andy McBrine over, but some more big hitting from the lower order in the remaining eight overs ??? David Miller (46 off 23) and Rilee Rossouw (61 off 30) ??? saw South Africa to 411-4 from their 50 overs, the second highest total in World Cup history.
Ireland started poorly in reply as a fired-up Steyn and Abbott shared five wickets between them.
Steyn had Paul Stirling out caught behind for nine ??? again only given after review ??? and accounted for Ed Joyce for a duck, edging to Amla at slip, with the century maker taking a smart catch as the ball was deflected by De Kock when diving in front of him.
At the other end Abbott picked up the wickets of Ireland captain Will Porterfield (12), Niall O???Brien (14) and Wilson, who also failed to trouble the scorers.
At 48-5 after only 10.2 overs, Andy Balbirnie ??? who batted nicely for 58 ??? and Kevin O???Brien (48) added 81 for the sixth wicket before Balbirnie fell, picking out Rossouw in the deep when attempting to pull Morkel over mid-wicket.
De Villiers picked up only his fourth ODI wicket with his part-time medium pace, getting Mooney to chop on to his stumps in the 31st over.
O???Brien fell two short of a half century as he departed in the same fashion as Balbirnie, the only difference being he provided Abbott his fourth wicket as Ireland slipped to 167-8.
Sorensen (22 off 19) provided some entertainment late on with three fours and a six to push Ireland past 200, but a ball after reaching the mark he perished, gloving a well-directed short ball from Morkel to De Kock.
Morkel then bowled Dockrell (25) to seal another impressive win for South Africa with five overs to spare, winning by 201 runs.
South Africa are next in action against Pakistan in Auckland on Saturday, while Ireland face India in Hamilton on Tuesday.