India took a 1-0 lead over South Africa by wrapping up a 113-run victory on the final day of the first Test at SuperSport Park.
The tourists had failed to win a Test in Centurion in their two previous matches, but bowled the Proteas out for 191 to draw first blood in the three-match series.
Battling knocks from Dean Elgar (77) and Temba Bavuma (35 not out) were in vain, with India sealing only their fourth Test win in South Africa just after lunch on Friday.
Bumrah got the breakthrough 10 overs into the morning session after the Proteas resumed on 94-4, trapping captain Elgar lbw with a ball that nipped back off the seam.
Quinton de Kock got a start but fell for 21 when he chopped on attempting to cut Siraj.
Bavuma continued to dig in, but India took lunch closing in on victory with South Africa 182-7 after Wiaan Mulder nicked a textbook probing delivery from Shami through to Rishabh Pant.
India only needed two overs of the afternoon session to finish the contest, Shami removing Marco Jansen before Ashwin got rid of Kagiso Rabada and had Lungi Ngidi caught by Cheteshwar Pujara at leg slip first ball.
Bumrah a tormentor on tour
India paceman Bumrah struck a massive early blow by seeing the back of South Africa skipper Elgar.
The dismissal of the gritty opener was Bumrah's 100th overseas in 23 Tests, making him the fastest to reach that landmark for India.
Bumrah produced a brilliant delivery to clean up Rassie van der Dussen on day four before also making a mess of Keshav Maharaj's stumps.
Battling Elgar and Bavuma show Proteas are up for the fight
South Africa have never lost a home Test series against India and although that record is seriously under threat, they have shown plenty of fight this week.
The Proteas have shown their potency with the ball and although they failed to make 200 in both innings, Elgar and Bavuma showed the sort of application that will be required if they are to mount a fightback.
India's relentless attack could not find a way to dismiss Bavuma, who faced 80 balls and batted for over two hours in the second innings, while Elgar led by example in a four-hour knock before falling to Bumrah.