The defending champions will need to bat brilliantly to avoid a third defeat in four games after watching a powerful Proteas line-up run riot after being put in by Jos Buttler at the Wankhede Stadium.
A revamped England side looked devoid of ideas in stifling humidity as their opponents brutalised them in the closing stages, taking 143 from a punishing last 10 overs.
Klaasen was in brutal mood, hammering a 61-ball ton and finishing with 109 in 67, while Marco Jansen made an unbeaten 75 from 42.
In all there were 13 sixes and 38 fours across the innings, which should have ended with England shipping 400 for the first time in their history only for South Africa to decline a second run off the final ball.
England’s previous worst day in the field came eight years ago at the Oval, where New Zealand hit 398 for five, and their have only ever been five bigger totals on the World Cup stage.
To win, and keep their ailing title defence alive, England will need to produce the third highest chase ever seen in one-day cricket.
David Miller was the Proteas' star with an unbeaten 59, as South Africa recovered from a dismal start in which they had slumped to 12-3.
The Netherlands had started poorly themselves with the bat, though Sybrand Engelbrecht steadied the ship with a 45-ball 40 before he was caught by Marco Jansen from Ottniel Baartman's delivery.
And with that, the Netherlands' scoring rate soon fizzled out, with the Dutch finishing on 103-9.
That lacklustre conclusion to their innings looked set to be swiftly forgotten as South Africa's top order disintegrated inside the first four overs, and the Netherlands seemed to be on their way to a famous win.
But a strong partnership of Tristan Stubbs (33) and Miller anchored South Africa's innings, and the Proteas emerged victorious with an over to spare.
Miller fittingly confirmed the victory with a six, smashing that shot through the leg side.
Sri Lanka remain bottom of Group D following their two-wicket defeat to Bangladesh earlier in the day.
Meanwhile, Afghanistan continued their excellent start at this year's tournament, beating New Zealand by 84 runs to move top of Group C.
Data Debrief: Dutch come unstuck
The defeat for the Netherlands ended their longest unbeaten streak in the competition, with the Dutch having won each of their last three matches at the T20 World Cup.
South Africa gained a measure of revenge, meanwhile, for their 13-run defeat to the Netherlands at the 2022 T20 World Cup.
Unicorns captain Corey Anderson won the toss and put the Freedom in to bat, a decision he was made to regret in the end as Washington piled up 207-5 from their 20 overs.
Captain Smith starred with a 52-ball 88 including seven fours and six sixes while Maxwell made 40 off 22 balls including one four and four sixes against 2-35 off four overs from Pat Cummins.
The Freedom then produced a dominant performance with the ball as well, dismissing the Unicorns for 111 in 16 overs to claim the title.
Tail-ender Carmi le Roux was the highest scorer in the innings with 20* as Rachin Ravindra and Marco Jansen took 3-23 from four overs and 3-28 from four overs, respectively.
Andrew Tye was also excellent with 2-12 from his two overs.
The Freedom are rightfully champions having suffered only one loss all season.
Full Scores:
Washington Freedom 207-5 off 20 overs (Steve Smith 88, Glenn Maxwell 40, Pat Cummins 2-35)
San Francisco Unicorns 111 all out off 16 overs (Carmi ls Roux 20*, Rachin Ravindra 3-23, Marco Jansen 3-28)
