Raut, who made 62 in a winning cause in the last match, continued her good form and scored 77 off 108 balls to anchor India's innings after the hosts were put in to bat. She struck 11 boundaries during her knock.
India, however, did not have the best of starts as they lost big-hitting opener Jemimah Rodrigues (0) in the second delivery of the innings when she nicked an away-swinging one to Sinalo Jafta off Shabnim Ismail (2/46).
But then Raut joined Smriti Mandhana (25 off 27) at the crease and the duo once again came to India's rescue, sharing 64 runs off 70 balls for the second wicket to take the team forward.
Mandhana, however, couldn't convert her start into a big score.
But skipper Mithali Raj, who became the first Indian woman to score 10,000 international runs on Friday, steadied the ship in Raut's company as the duo stitched 77 runs for the third wicket.
Mithali, only the second internationally to complete 10,000 runs across all formats, departed soon after reaching the major milestone, flicking a Anne Bosch delivery straight to Mignon du Preez at mid-wicket.
She made 36 runs off 50 balls and hit five boundaries in the process.
Raut also fell soon after and Harmanpreet Kaur (36 off 46) played a quick knock before becoming Ismail's second victim as India reached 221 for five in the 45th over.
Towards the end, Deepti Sharma (36 not out) and Sushma Verma (14 not out) remained unbeaten and shared 27 runs for the sixth wicket to take India close to the 250-run mark.