Thanks to Taijul, who became only the second Bangladeshi bowler after Shakib Al Hasan to reach 200 wickets, Bangladesh reduced South Africa to 6-140 before bad light brought a premature end to Monday's play in Mirpur.
South Africa lead by 34 runs after a day when 16 wickets fell, with Kyle Verreynne on 18 with Wiaan Mulder on 17.
With the pitch showing signs of deteriorating, Bangladesh sensed a chance to deny South Africa a big lead.
"To be honest, we did not get a good total. If we had scored around 220 or 250, the match would have been in our hands," Taijul said. "But we have to do really well from here. We should play better in the next innings."
Mulder (3-22), Kagiso Rabada (3-26) and Keshav Maharaj (3-34) earlier shared nine wickets among them to wreak havoc on Bangladesh after their captain, Najmul Hossain Shanto, won the toss and opted to bat first.
Mulder took a wicket in each of his first three overs to put Bangladesh in early trouble.
The rot began with the dismissal of Shadman Islam, who played a needless shot against a delivery that moved away to edge to second slip for a duck.
Mominul Haque was caught behind, while Shanto flicked straight to short mid-off, leaving Bangladesh on a precarious 3-21 in the sixth over.
Rabada then got in on the act, rattling the stumps of veteran Mushfiqur Rahim (11) for his 300th wicket in Test cricket. In doing so, he became the sixth South African bowler to achieve the feat.
"When I came on to bowl this morning, I wasn't really thinking about that final wicket," Rabada said. "I was more focused on how we were going to win this Test, especially after losing the toss and bowling first. But when it happened, it was just a relief. Everyone plays for milestones."
The situation went from bad to worse for the hosts as Rabada got the better of Liton Das, caught at gully for one.
Kagiso Rabada has become the sixth South African bowler to take 300 Test wickets. (Dean Lewins/AAP PHOTOS)
On the stroke of lunch, left-arm spinner Keshav Maharaj trapped Mehidy Hasan lbw for 13.
Opener Mahmudul Hasan, who battled throughout the morning session, was the first victim after lunch.
He made 30 off 97, with two fours and one six, before Dane Piedt bowled him out for his only wicket in the innings.
Maharaj then had debutant Jaker Ali Anik stumped for 2 as Bangladesh were reduced to 8-76 and in danger of being all out for less than 100, but Taijul's 16 runs helped his side into three figures.
Taijul then took the responsibility to inject life into the contest.
Hasan Mahmud got the better of stand-in captain Aiden Markram in the first over before Taijul ripped through the middle order.
Tristan Stubbs and Tony de Zorzi shared 44 runs for the second wicket as South Africa appeared to take the full control. But Taijul intervened as he claimed the wicket of Stubbs, who made 23 off 27 but was done completely by the turn.
Taijul then baffled David Bedingham and Tony de Zorzi, who top-scored with 30, with extra bounce before taking the wicket of Matthew Breetzke, his 200th in Test cricket. He also had Ryan Rickelton caught for 27.
Verreynne and Mulder saw out the day, sharing 32 runs in an unbeaten seventh-wicket stand.
