Left-arm quick Starc sent back skipper Temba Bavuma for a fourth-ball duck in the first over and soon Hazlewood had the in-form Quinton de Kock caught for three.
The new-ball bowlers kept up the attack with the wickets of Aiden Markram (10) and Rassie van der Dussen (six) to reduce South Africa to 24-4 after the Proteas elected to bat under cloudy skies.
Heinrich Klaasen (10) and David Miller (10) had taken the score to 44-4 in the 14th over when rain sent the players off the field at Eden Gardens.
If the action falls victim to the weather, then there is a reserve day set aside on Friday.
Should no result be possible, South Africa would be declared the winners as they finished higher than Australia in the group phase.
South Africa are chasing their first ever World Cup final after faltering in four semi-finals including two against Australia in 1999 and 2007.
The winners will take on undefeated hosts India, who knocked out New Zealand in the first semi-final, in Sunday's final at Ahmedabad.
