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Sri Lanka now play Australia in the second semi-final on Wednesday with giant-killers Pakistan playing the West Indies in the first semi-final on Monday.
Australia qualified with a 62-run win over New Zealand on Sunday, a day after Pakistan knocked out defending champions India with a two-wicket victory and the West Indies beat England by 18 runs.
Man-of-the-match Rajapaksa belted 10 fours as he top-scored for Sri Lanka with 79 off 78 balls and put on 91 in 16 overs for the third wicket with fellow left-hander Rumesh Buddika at Lincoln Green.
Danishka Gunathillike added an unbeaten 24 from 16 balls at the tail of the innings as Sri Lanka posted 293 for eight from their 50 overs.
South Africa were two wickets down by the seventh over, when rain stopped play and a revised target of 292 in 49 overs was set.
After play resumed Josh Richards (46) and Colin Ackerman (21) put on 50 in 56 balls for the third wicket.
But the fall of Ackerman in the 16th over triggered a collapse from which South Africa never recovered and they were all out for 145 after 36.4 overs with medium-fast Charith Jayampathi taking 4-31.
At Rangiora, Australia batted first against New Zealand and scored 232 for eight with skipper Mitchell Marsh, son of former Australian wicketkeeper Rodney Marsh, top scoring with 53.
New Zealand launched a cautious reply, reaching 26 without loss in 10 overs before Jason Floras made the breakthrough when he bowled Harry Boam who had made just six from 27 balls faced.
New Zealand were eventually all out for 170 in the 44th over, 62 runs short of their target.
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