The Black Caps, chasing a target of 171 runs on Monday, needed just seven runs from the final over with five wickets in hand and big-hitting Ross Taylor at the crease, unbeaten on 62.
But Steyn had Luke Ronchi caught off his first ball and, despite being hit for four, took the wicket of Nathan McCullum with his fifth delivery, leaving Taylor needing to score three off the final ball of the Super 10 match to win.
The South African bowled a full-pitched delivery to Taylor who could only drive it back to Steyn, who ran him out at the non-striker’s end, consigning New Zealand to a two-run defeat.
“Steyn proved why he’s been the world’s best bowler for such a long period of time,” said Du Plessis. “It was a fantastic bowling performance.”
“As a captain, to have a guy like that in the team anything is possible,” he added.
“I thought New Zealand got themselves in the position where they should have won the game and something special was going to be required for us to take it away from them and Dale did just that.”
Jean-Paul Duminy’s unbeaten 86 had earlier lifted South Africa to 170 for six in their 20 overs.
New Zealand skipper Brendon McCullum hailed Steyn as a top bowler.
“We probably should have won it easily, can’t take it to the final over against a top bowler like Steyn,” said McCullum, whose dismissal for four off leg-spinner Imran Tahir was a big setback for New Zealand.
“If we had a little bit of luck, we’d have got across the line.”
South Africa play the Netherlands in their next Group one fixture on Thursday while England take on Sri Lanka.
Defending champions West Indies face Bangladesh in a Group Two match in Dhaka on Tuesday.
The top two teams from each group qualify for the semi-finals. The final will take place in Dhaka on April 6.