Mahmudullah made 67 while Tamim added a patient 57 as Bangladesh reduced their deficit to 69 runs after the top-ranked Proteas were dismissed for a modest 248.
Vernon Philander trapped Mahmudullah leg-before just before rain ended play in the third session, swinging the game back towards the visitors somewhat after lowly Bangladesh appeared to have taken the advantage.
Mahmudullah, who hit 10 fours off 138 balls, put on 89 with Tamim for the third wicket, providing the backbone of the Bangladesh innings after the hosts resumed on Wednesday on 7-0.
Tamim and fellow opener Imrul Kayes mixed aggression with caution in the morning to blunt the new ball, scoring 41 in the first hour.
Occasional medium pacer Stiaan Van Zyl dragged Imrul out of his crease for Quinton de Kock to complete a stumping, breaking a resolute 46-run opening stand.
Imrul departed for 26 after leading the initial charge with three fours.
Spinner Simon Harmer then took out Mominul Haque's off stump to leave Bangladesh struggling on 55-2 before Tamim and Mahmudullah rebuilt the innings.
Shakib Al Hasan, one not out, will resume with his skipper on Thursday as Bangladesh push for an unprecedented Test victory against the Proteas.
Bangladesh, who have a dismal Test record, had made an impressive start to the match on the opening day on Tuesday.
Paceman Mustafizur Rahman claimed 4-37 in his Test debut and leg-spinner Jubair Hossain added 3-53 as Bangladesh bowled out the visitors for 248.
The second match of the two-Test series will be played in Dhaka from July 30-August 3.
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