Chasing 120, Mithali top scored a 66-ball unbeaten 51 and Mona Meshram contributed with a patient 93-ball 38 to overhaul the target in just 41 overs after Jhulan (3/20) and Shikha Pandey (3/22) made life difficult for the home side.
In the process, Goswami also became the leading wickettaker in women's Ond-day Internationals, eclipsing Australian Cathryn Fitzpatrick, and taking her haul to 181 wickets from 153 outings.
Earlier, Mignon du Preez (38) and wicketkeeper-bat Trisha Chetty (28) were among the four South Africans to manage the double figures along with Marizanne Kapp (23) and Laura Wolvaardt (17) as the Indian opening bowling duo of Jhulan and Shikha gave little away.
India will now face Zimbabwe in their next match on Thursday.
Brief Scores: South Africa 119 (Mignon du Preez 38, Trisha Chetty 28, Jhulan Goswami 3/20, Shikha Sharma 3/22) lose to India 121/3 (Mithali Raj 51 not out, Mona Meshram 38, Shabnim Ismail 2/17) by 7 wickets.