India picked a four-pronged spin attack for the series opener against the world champions, but it was Bumrah's 5-27 that wrecked South Africa, who were 57 for no loss at one stage.
India were 37 for one at stumps with KL Rahul (13) and Washington Sundar (six) in the middle and a lengthy batting lineup waiting in the wings.
South Africa will have their work cut out with the ball, especially as they are without the leader of their pace attack, Kagiso Rabada, who misses the match with a rib injury.
Implosion for South Africa
There was no inkling of South Africa's impending implosion when Aiden Markram (31) and Ryan Rickelton (23) combined in a half-century opening following skipper Temba Bavuma's decision to bat after winning the toss.
The Proteas began cautiously, and Markram needed 23 balls to get off the mark, which he did with a straight driven four off Mohammed Siraj, before counter-attacking.
India captain Shubman Gill introduced spin in the eighth over, but Markram hit Axar Patel for two fours in that over before smacking a six in the left-arm spinner's next over.
Bumrah pegged back Rickelton's off-stump to break the stand and dismissed Markram with Rishabh Pant, returning from a foot injury, taking a diving catch behind the stumps.
Dhruv Jurel took a smart catch to remove Bavuma for three after the South Africa skipper deflected a Kuldeep Yadav delivery into the fielder's hands at leg slip.
India in control
With Bumrah leading the charge, India prised out five wickets in the second session to snuff out South Africa's hopes of reaching the 200 mark.
Wiaan Mulder (24) was dismissed by Kuldeep while Tony de Zorzi (24) also failed to convert the start - both batters wasting a review each in their bid to reverse lbw decisions.
Siraj then chipped in by dismissing Kyle Verreynne and Marco Jansen in a four-ball burst to flatten South Africa's middle order.
Bumrah, cheered on by a 36,000-plus crowd, returned to send down a two-wicket over to hasten South Africa's collapse soon after the tea break.
India suffered an early blow in their reply when Yashasvi Jaiswal (12) chopped a Marco Jansen delivery onto his stumps, but Rahul and Sundar saw through the rest of the day's play.
