England were bowled out for 179 in 38.2 overs after their departing skipper Jos Buttler had chosen to bat, and South Africa before reaching their target with a huge 125 balls to spare following a 127-run third-wicket stand between Klaasen (64) and Van der Dussen (72no).
The win means the Proteas top group B and join Australia in the semi-finals, with the fixture line-up only to be confirmed when the already-qualified India and New Zealand meet in Dubai on Sunday to battle for top spot in group A.
The match was Buttler's last in charge for England, as he's stepping down from his role, with his side having ended up losing all three pool fixtures in the tournament.
England were already out of contention going into Saturday's game and, after winning the toss, their innings never got going as they played some poor shots on a good batting wicket.
"A really disappointing performance," Buttler said. "We were so far short of the mark there today. It was a decent surface -- pretty slow and a bit two-paced.
"We're not going on and making those big, telling contributions which has been a story of this side for some time now with the bat."
South Africa did well to expose England's desire to chase the run-rate with injudicious shots but also bowled both sides of the wicket and will need to tighten up their lines, whoever they face in the semi-finals.
"The wicket was quite slow, slower than we expected," South Africa stand-in captain Aiden Markram said. "The boys adapted really well and kept trying to hold a length for as long as they could. We took wickets throughout."
Joe Root top-scored for England with a well-constructed 37 before he was bowled by seamer Wiaan Mulder (3-25), while Buttler managed 21.
There was little else to enthuse over for the side, with Ben Duckett (24) and Jofra Archer (25) also managing starts.
South Africa lost Tristan Stubbs (0) early in their chase, as he played on to his wicket from a delivery by Archer (2-55).
But once Klaasen and van der Dussen got together, there was never much doubt about the result.
Van der Dussen's unbeaten 72 came from 87 balls and included three sixes as he anchored the innings along with the more explosive Klaasen, whose 64 came off 56 balls, with 11 boundaries.
But South Africa will sweat over the fitness of Markram, who left the field with a hamstring injury, ahead of their semi-final, while captain Temba Bavuma and opening batter Tony de Zorzi missed the game through illness.
