After limiting Bangladesh to just 106-3 in 20 overs at Dubai, South Africa reached their target with 16 balls to spare to win by seven wickets.
With three wins and one defeat, they sit top of Group B with six points.
However, 2009 champions England, with two wins so far and with two games still to play, are likely to secure one of the semi-final spots.
That would leave South Africa and West Indies vying for the last place.
West Indies, the 2016 champions, have four points but a better net run-rate than South Africa although the Caribbean side have to face England in their concluding pool game on Tuesday.
Earlier in Sharjah, New Zealand trounced winless Sri Lanka by eight wickets, chasing down a 116-run target with 15 balls to spare.
New Zealand have four points from three games but remain third in Group A behind India, also on four points, and defending champions Australia who have a perfect six points from three outings.
India face six-time winners Australia on Sunday before New Zealand complete their group campaign against Pakistan on Monday.
Player of the match Tazmin Brits steered 2023 runners-up South Africa to victory with a steady 42 off 41 balls, an innings which featured five fours.
Anneke Bosch made a run-a-ball 25 before Marizanne Kapp (13 not out) and Chloe Tryon (14 not out) saw their side home.
"The way we started with the ball was amazing. Would have loved to chase it down faster and that was a message that went out, but we didn't get there," said South Africa captain Laura Wolvaardt who bemoaned her side conceding 13 extras when Bangladesh batted.
Bangladesh won the toss and chose to bat but lost opener Dilara Akter off the second ball for nought.
The Asian side struggled to make any headway against the South African attack.
Sobhana Mostary top-scored but her 38 took 43 balls with skipper Nigar Sultana making an undefeated 32 from 38 deliveries.
Kapp claimed 1-10 from her four overs of medium pace while Nonkululeko Mlaba finished with 1-11.
Off-break bowler Mlaba has enjoyed an impressive tournament with the ball and has nine wickets from four games, making her the leading wicket-taker.
"There's a lot of intent and emotion there," said Bangladesh captain Sultana after a tournament which brought them just one victory -- an opening day win against Scotland.
Plimmer shines for Kiwis
In Sharjah, New Zealand limited Asian champions Sri Lanka to 115-5 in 20 overs.
Georgia Plimmer, a 20-year-old opener, then hit 53 before captain Sophie Devine smashed the only six of the match to take her team to 118 for two with 15 balls left.
Devine said the White Ferns did not discuss run rate before starting their chase.
"We know that winning the game was the most important thing," she said.
"We will see how results play out tomorrow and we have our final game against Pakistan and by that point we'll know exactly where we stand and what's going to be required."
Captain Chamari Athapaththu opened and made the top score of 35 as her Sri Lanka team passed 100 for the first time in the tournament.
"We improved a little bit today but we needed another 20-25 runs," said Athapaththu.
Despite scoring at almost a run a ball, Plimmer hit only four fours and had to run for most of her 53 on a day when temperatures hit 39 degrees Celsius.
Amelia Kerr, who hit 34 to follow her two wickets, and Devine, who made 13, sped New Zealand to victory with a late flurry of boundaries.
