South Africa resumed on 247-6, with Muthusamy on 56 and Kyle Verreynne on 38, extending their unbeaten seventh-wicket partnership to 70 by the end of the first session.
Play started 30 minutes earlier than usual due to early sunsets, with session breaks reordered so that the shorter tea break came before lunch.
The tourists, who won the toss and elected to bat on Saturday, are aiming for their first series victory in India in 25 years after winning the opener of the two-match series.
Verreynne hit a boundary to take South Africa past 300, while left-hander Muthusamy reached his fifty with a single off Ravindra Jadeja, looking in complete control.
Earlier, Tristan Stubbs fell for 49 and skipper Temba Bavuma scored 41, sharing an 84-run partnership for the third wicket on day one—the only fifty-plus scores before Muthusamy’s knock.
Muthusamy and Verreynne used their feet to tackle spinners in a sedate first hour of play when only 28 runs were scored, including two fours.
Muthusamy had a reprieve on 48 when left-arm spinner Jadeja had him given out lbw but the batsman's review showed the a muurmur when the ball passed his glove and the decision was overturned.