The hosts were bowled out for 159 in the second session and were quickly put back in to bat by the Proteas. The Proteas had posted an impressive total of 575-6 in their first innings, giving them a commanding lead in the two-match series, which they now lead 1-0.
Home skipper Najmul Hossain Shanto was alone at the crease on 13 after Zakir Hasan was trapped for a stumping off Senuran Muthusamy in the last ball before the break for seven.
Dane Paterson, in his first over, sent opener Shadman Islam back for six off a good-length delivery from round the wicket, edged to wicketkeeper Kyle Verreynne.
Muthusamy also struck off his first over when his tossed-up delivery to Mahmudul Hasan Joy kissed the bat and landed in the palms of Aiden Markram at slip.
Bangladesh fell into further trouble when Keshav Maharaj reduced them to 29-3 to send back Mominul Haque, who scored 82 in the first innings, for a duck.
Fast bowler Kagiso Rabada finished the first innings with 5-37 for South Africa.
The hosts were reduced to 48-8 on day three in their reply before a mini-fightback by Mominul.
Bangladesh resumed at 38-4 but Rabada struck in the fourth over with a cracking delivery that Najmul, on nine, could only edge to Verreynne.
Mushfiqur Rahim went without scoring when Paterson induced him to pop up a simple catch to Tony de Zorzi at short leg to leave Bangladesh 47-6.
That soon became 48-8 when Rabada struck twice in one over as Mehidy Hasan Miraz was caught behind and debutant Mahidul Islam Ankon offered no shot to a ball nipping back and was trapped lbw for a duck.
The burst gave Rabada his second five-wicket haul of the series, to go with his 6-46 in the first Test victory in Mirpur.
South Africa had three batsmen hit maiden Test centuries in their first-innings 575 De Zorzi , Tristan Stubbs and Wiaan Mulder .
The tourists are looking to sweep the two-match series after winning the first Test by seven wickets.
Bangladesh have never won a Test against South Africa in 15 attempts.
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