Making a comeback after more than seven months, Klaasen never took his foot off the pedal and got his team out of the woods after Bhuvneshwar pegged them back to 29/3 inside the powerplay overs.
Rassie van der Dussen smashed 75 off 46 balls while David Miller scored 64 off 31 balls to guide Proteas to a historical run chase.
Skipper Temba Bavuma hit a restrained 110 off 143 balls, while Rassie van der Dussen blasted an entertaining 129 not out off 96 balls
Resuming the day 101 for two, South Africa go the job done in 63.3 overs with major contributions coming from Keegan Petersen (82), Rassie van der Dussen (41 not out) and Temba Bavuma (32 not out).
The 54-run stand between Petersen, who had 10 boundaries in his innings and Van Der Dussen, who played 72 balls, made it easier for the Proteas.
Kagiso Rabada sent back England's Chris Woakes, skipper Eoin Morgan and Chris Jordan for the tournament's third hat-trick.
South Africa now need to restrict England to 131 runs to qualify for the semifinals.