The wicketkeeper's 114, his second Test ton, pushed South Africa to 308 and a 202-run lead in the first innings in Mirpur.
When bad light forced stumps on day two on Tuesday, Bangladesh were 3-101 and still needing 101 more runs to make South Africa bat a second time.
Opener Mahmudul Hasan was 38 not out and Mushfiqur Rahim 31 in an unbeaten 42-run stand for the fourth wicket.
Verreynne shared two vital partnerships - 119 with Wiaan Mulder for the seventh wicket, and 66 with Dane Piedt for the ninth.
He was the last batter out after hitting eight fours and two sixes in his 144-ball knock.
Mulder made 54 off 112 ball and Piedt scored 32 off 87 deliveries.
South Africa started the day at 6-140 and in danger of being bowled out without a significant lead as the Bangladesh spinners continued enjoying turn and bounce since the day-one evening session.
But Verreynne and Mulder batted resolutely to quell the tension. They played sweep and reverse-sweep shots perfectly to astutely deal with the spinners and take control.
"I don't think there's been any extra emphasis (on the sweep shot)," Verreynne said.
"It is something that I have worked on quite a bit against spinners. It is not really a subcontinent specific thing. I would play it back home against spin as much as possible."
Verreynne raised his fourth Test half-century from 67 balls by sweeping Taijul Islam to short square leg, and took on the other spinners Mehidy Hasan Miraz and Nayeem Hasan.
Mulder reverse-swept offspinner Nayeem for four through third man to reach his maiden half-century off 105 balls.
With the partnership growing, pacer Hasan Mahmud gave the home side some respite with two wickets in consecutive deliveries.
He had Mulder caught by Shadman Islam at wide slip with a back-of-length delivery, and clean-bowled Keshav Maharaj for a duck.
Bangladesh looked to wrap up the innings with South Africa ahead by 121 runs, but Piedt gave Verreynne ample support to rotate the scoreboard rapidly and lift the lead past 200.
Verreynne brought up his century off 134 balls, sweeping Taijul powerfully through deep square leg for a single.
"It is definitely the best innings of my Test career," Verreynne said. "It is probably the toughest conditions I have played under in terms of heat and humidity."
Miraz (2-63) broke through with the wicket of Piedt, trapped lbw for 32.
Verreynne hit two sixes off Miraz before the bowler got the better of him to finally close South Africa's innings.
Hasan Mahmud took 3-66, while Taijul couldn't add any wickets to his overnight tally and finished with 5-93.
Fast bowler Kagiso Rabada took two wickets in three balls to reduce Bangladesh to 2-4 in the third over.
Mahmudul Hasan and captain Najmul Hossain Shanto resisted to add 55 runs, before Keshav Maharaj dismissed the latter for 23.
Hasan and Mushfiqur Rahim then reached stumps without further trouble.
"If we can set them a 200-run target, we'll win the match," fast bowler Hasan Mahmud said.
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