While Tamil Nadu left-hander Mukund contributed 72 runs, which is the top score so far, skipper Rayudu played an enterprising innings with his 71 coming off 81 deliveries, including eight fours and three sixes.
Mukund's fellow opener Jiwanjot Singh also chipped in with a handy 52 at the start of the innings.
Young wicketkeeper-batsman Ankush Bains partially made amends for conceding 24 byes in the South African innings as he was unbeaten on 34 at the end of the day's play. Giving him company at the other end was Axar Patel on 16.
India began their innings on a cautious note but Mukund and Jiwanjot added 96 runs for the first wicket in 28 overs.
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It was left-arm seamer Lonwabo Tsotsobe, who angled one across to Jiwanjot, who nicked it to Dane Vilas behind the stumps.
Mukund, who again showed a lot of patience during his 136-ball knock with 13 fours, was joined by Baba Aparajith (34) and the duo added 72 for the second wicket.
In his very next over, Piedt also got rid of Aparajith, who provided a catch for Omphile Ramela.
At 169 for three, it was Rayudu and Sheldon Jackson (25) at the crease. The duo added 55 runs for the fourth wicket before Piedt got his third wicket, cleaning up Jackson.
However, Rayudu started taking on Piedt at the other end with Vijay Shankar (21) for company. Tsotsobe got his second wicket as Shankar edged one to Vilas.
Rayudu along with Bains added 56 runs in 14.3 overs before Piedt finally got rewards for flighting his deliveries with Temba Bavuma holding onto Rayudu's catch.