JOIN USSri Lanka now play Australia in the second semifinal on Wednesday while giant-killers Pakistan take on West Indies in the first semifinal on Monday.
Australia qualified with a 62-run win over New Zealand.
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 four for 31.
At Rangiora, Australia batted first against New Zealand and scored 232 for eight with skipper Mitchell Marsh top-scoring with 53.
New Zealand were all out for 170 in the 44th over, 62 runs short of their target.
Brief scores: Australia: 232/8 in 50 overs (Nic Maddinson 37, Mitchell Marsh 53, Alex Keath 32, Tim Armstrong 39) bt New Zealand: 170 all out in 43.2 overs (Tom Latham 25, Craig Cachopa 34, Doug Bracewell 33; Josh Hazlewood 2-24, Luke Doran 2-36, Jason Floros 2-27, Nick Buchanan 2-29).
Sri Lanka: 293/8 in 50 overs (Bhanuka Rajapaksa 79, Rumesh Buddika 42, Kithuruwan Vithanage 45, Akshu Fernando 28; Jerry Nqolo 2-46) bt South Africa: 145 all out in 36.4 overs (Josh Richards 43; Charith Jayampathi 4-31, Nadeera Rajaguru 2-15).