Pakistan resumed their second innings needing a further 228 for victory with seven wickets in hand, but their chances faded when Olivier removed the risky Babar Azam (21) and captain Sarfraz Ahmed (0) in consecutive balls.
Unless Test matches are added to the worldwide schedule India will end as the number one Test team in the world at the annual cut-off in April. They played really well.
Steyn's two wickets took his total in Tests to 433, bringing him level with Sri Lanka's Rangana Herath and Stuart Broad of England in joint eighth place on the all-time Test wicket-takers list.
"Our fast bowlers know what they want to do, which makes your life easier as a captain".
"I'm extremely proud, a 3-0 series win is just what we asked the guys for", stand-in South Africa captain Dean Elgar said at the post-match presentation. "They have worked out their game-plans and know how to execute", he added.
Oliver opened the door in the third over of the day when he had the risky Babar Azam (21) hopping with the vicious bouncer that the dynamic stroke-maker could only fend behind to Quinton de Kock. "We are very fortunate to have that arsenal of four fast bowlers".
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That was incredible, thank you so, so much to everyone that came out tonight", he said after being given a rapturous send-off. I'm still in pain doing just basic day-to-day things. "Amazing.
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But Philander then had the batsman caught at slip as Pakistan were dismissed for 273, Aiden Markram's cracking diving catch at gully to remove Faheem Ashraf off Kagiso Rabada (3-75) a highlight.
Hasan Ali (22) had some fun - a six over midwicket his best moment - before he was caught and bowled by Rabada, while the match ended in comical fashion when a mix-up between Shadab and Mohammad Abbas (9) saw the latter run out by Markram shortly before lunch.
Olivier took 24 wickets in the three-Test series at an average of 14.71 and he made the crucial breakthrough early on Monday.
Nightwatchman Mohammad Abbas (11) was also dropped twice off fairly simple chances to battle through the first hour and frustrate the home side, though South Africa's luck changed with two wickets in one Duanne Olivier (2-20) over.
"Shan Masood played really well (with the bat) and Babar showed his class. Shan and Babar were the positives (from the series)".
Shafiq edged Vernon Philander behind for 65 and with him went Pakistan's hopes of winning the Test match.