Chasing 322 for victory, South Africa were 211 for four with 13 overs left when the rain came, leaving them stranded at seven runs behind the Duckworth-Lewis (DL) par score.
New Zealand, the 2019 runners-up, won the toss and opted to bat at the Greenfield International Stadium in Thiruvananthapuram, with opener Conway hitting a 73-ball 78 to lay the platform for the Blackcaps.
Captain Kane Williamson, who has been ruled out of Thursday's tournament opener against England to give him more time to complete his rehabilitation from knee surgery, also contributed 37 runs.
Conway and Williamson then left the field retired hurt, but both retirements appeared to be precautionary. Lungi Ngidi and Marco Jansen picked up three wickets apiece as New Zealand ended on 321-6.
In the chase, Quinton de Kock was the cornerstone of South Africa's innings and was involved in significant partnerships with Rassie van der Dussen (51) and Heinrich Klaasen (39).
With De Kock looking sharp and still batting on 84 not out when the match was interrupted, South Africa will likely feel aggrieved at the outcome. The Proteas face Sri Lanka in their World Cup opener on Saturday.
England beat Bangladesh
Moeen Ali smashed a quickfire half-century to guide champions England to a four-wicket victory over Bangladesh in a rain-affected World Cup warm-up game in Guwahati on Monday.
Chasing a revised target of 197 in 37 overs following a lengthy rain interruption, England were hurried along by Moeen's blistering knock of 56 off 39 balls, with the all-rounder hitting two fours and six sixes.
Jonny Bairstow (34), Joe Root (26 not out) and captain Jos Buttler (30) also made telling contributions, while Bangladesh's Mustafizur Rahman picked up two early wickets, dismissing both openers.
Earlier, Bangladesh won the toss and opted to bat, with over half of their 188 runs coming through Tanzid Hasan (45) and Mehidy Hasan Miraz (74).
England, keen to quickly acclimatise to the conditions after their first warm-up match against hosts India was washed out, employed nine different bowlers, five of whom took wickets, but Reece Topley stood out with figures of 3-23 in five overs.
"Very valuable, obviously didn't have the first game," Moeen said. "More important for the bowlers, invaluable time in the field.
"I just want to play how I can play. Sometimes you have to adapt. Play to your strengths. Felt like I played well so wanted to keep the momentum.
"We're ready (for the opener), got enough time. Going to be a massive game, NZ are a dangerous side, but we're ready."
Bangladesh take on Afghanistan in their first match of the World Cup on October 7th.
