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.
The verdict comes nearly two months after the tournament, which India won by beating South Africa by seven runs in the final on June 29.
In those two games at the Nassau County Stadium, Sri Lanka were skittled out for 77 against South Africa on June 3 while Ireland were dismissed for 96 by India two days later.
In the second game, a number of players from both sides copped were struck on the body as a result of the uneven bounce in the pitch.
Indian captain Rohit Sharma had to retire hurt as a precautionary measure after being hit on the upper arm by Josh Little.
India wicketkeeper Rishabh Pant and Ireland batsman Harry Tector were also struck as a result of the nature of the surface.
At the time, Andy Flower had said the pitch in New York was "bordering on dangerous," while former England captain Michael Vaughan called it "shocking."
The venue in New York was readied in five months. Damian Hough, chief curator at the Adelaide Oval, was brought in by the ICC to prepare the drop-in pitches.
But after criticism from various quarters, the ICC had acknowledged that the pitches there had been substandard and the short turnaround time between matches didn't help. New York hosted eight games in two weeks.
Remedial work was carried out, and ahead of the following game, between Canada and Ireland, areas where grass shoots were growing under the cracks were covered with topsoil and rolled in to make the surface much flatter.
The surfaces used for the subsequent fixtures in New York, including India vs Pakistan, where 119 beat 113 for 7, have been rated as "satisfactory".
The bigger point of scrutiny, perhaps more than New York, was on the surface for the semi-final, where Afghanistan were bowled out for 56 at the Brian Lara Cricket Academy.
The wicket was wildly unpredictable as some deliveries rolled along the ground while a few reared up from a similar spot.
At the time, Afghanistan head coach Jonathan Trott was clear "that's not a pitch you want to play a semi-final on".
Generally, surfaces at the Brian Lara Cricket Academy were a challenge. West Indies, who scored 149 there and defended it narrowly against New Zealand, were tottering at 30 for 5 at one stage.
The ICC rates pitches, and outfields, for all international games on a scale of very good to unfit: very good, good, satisfactory, unsatisfactory and unfit.
The surface at Providence for the India vs England semi-final was rated "satisfactory", while the Kensington Oval deck for the final between India and South Africa was rated "very good".
The ICC was largely happy with the state of the outfields, with only New York and Guyana receiving "satisfactory" ratings, while the others were "very good".
India captain Rohit retired hurt after making 52 in the second innings in New York as Rahul Dravid's side chased a 97-run target with more than seven overs to spare.
The opening batter was struck on the upper arm by a vicious Josh Little delivery in the ninth over, subsequently leaving the field before Rishabh Pant's 36 not out eased India over the line.
Rohit conceded the pain remained in his post-match interview before focusing on the positives of his team's bowling showing, with Hardik Pandya, Jasprit Bumrah and Arshdeep Singh all impressing.
He said: "Just a little sore. New ground, new venue, wanted to see what it's like to play on; I don't think the pitch settled down, there was enough there for the bowlers.
"Stick to your basics, think about Test match bowling. Arshdeep can swing the ball into the right-handers and that set the tone.
"If the conditions are there for the seamers, we wanted them in the squad. The spinners will play their part later in the tournament. We are open to making changes to the team's needs."
Bumrah produced a blistering spell, taking two wickets for just six runs from his three overs as Ireland limped to 96 all out in New York.
"When you come here and the ball is seaming around with some bounce and pace, I would never complain," Bumrah added. "You have to be proactive, you can't preempt things.
"You realise how the wicket is and then go back to what works for you. Once the seam goes down the pitch does settle down.
"You have to be prepared to bowl in all conditions, so very happy today."
India meet fierce rivals Pakistan on Sunday after their opening World Cup victory, and Rohit expects his team to be flexible once again dependent on conditions.
"I don't know what to expect from the pitch but we will prepare as if conditions are going to be like that," Rohit continued.
"That will be a game where all 11 of us need to come together and contribute.
"It was scratchy but good to spend time in the middle, hopefully we can do the same [against Pakistan]."
