The Proteas were chasing a reduced target of 64 in seven overs and were well-placed on 51 without loss after three overs before rain came down again and the umpires took the decision to call the match off.
After a delayed start the match at the Blundstone Arena had initially been reduced to a nine-over contest, Zimbabwe reached 79-5.
Wessly Madhevere top-scored with an unbeaten 35 off 18 balls after Zimbabwe did well to recover somewhat after losing four wickets with 19 runs on the scoreboard.
Paceman Lungi Ngidi took two wickets while fellow seamers Wayne Parnell and Anrich Nortje took a scalp each as the Proteas restricted their African neighbours.
With the threat of rain always in the air, South Africa raced out of the blocks and Quinton de Kock hit four fours and a six in the opening over off Tendai Chatara.
The Proteas wicketkeeper-batter did everything he could to win a race against the elements, clubbing a total of eight fours and a six as he made 47 not out in just 18 balls but it was not enough.
With a minimum of five overs required to ratify the result the points were shared and, not for the first time, South Africa were frustrated by the weather at a major tournament.