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The Kiwis were home and dry with more than seven overs to notch up their third successive win and go top of the table.
Williamson missed wins over defending champions England and the Netherlands to work on his fielding and running between the wickets.
The Black Caps again demonstrated their knack of punching above their weight in showpiece events.
In terms of buildup India could not have asked for better, winning the Asia Cup last month and beating Australia 2-1 in a subsequent ODI home series.
Chasing 322 for victory, South Africa were 211/4 with 13 overs left when the rain came, leaving them stranded at seven runs behind the Duckworth-Lewis (DL) par score at the Greenfield International Stadium in Thiruvananthapuram.
It is imperative the sides to preserve their energies and peak at the correct time to stand a chance of making it to the final.
The two-time runners-up will take on South Africa in a practice match at the Greenfield International Stadium on Monday.
New Zealand chased down the 346-run victory target set by Pakistan with 6.2 overs to spare.
Tom Latham is set to lead New Zealand in what would be a rematch of the 2019 final.
Liverpool won 3-1 while Arsenal were held 2-2 by Spurs.