Paceman Steyn claimed 3-30 while off-spinner Duminy took 3-27, as a mini collapse in the final session restricted Bangladesh to 246-8 at stumps.
Skipper Mushfiqur Rahim scored the lone half-century for Bangladesh and made a 94-run partnership with Mahmudullah for the fourth wicket after he won the toss and elected to bat.
Rahim hit 65 off 125 balls, his 15th Test fifty and first in 12 innings as Bangladesh made slow progress before South Africa took control of the game in the final session at the Sher-e-Bangla National Stadium.
The 32-year-old Steyn almost reached the landmark in his second over of the day but Dean Elgar dropped Tamim at second slip.
Skipper Hashim Amla made no mistake in Steyn's next over when he took a chest-high catch at first slip.
Shaun Pollock is the only other South African to pass the 400-mark, taking 421 wickets before retiring in 2008.
Steyn, who is playing his 80th Test, became the 13th cricketer to take 400 or more wickets in Test matches.
He is one of only three currently active cricketers to achieve the feat, joining India spinner Harbhajan Singh and England paceman James Anderson.
Duminy forced an edge from Mominul for 40, ending the batsman's 69-run second wicket stand with Kayes.
Kayes followed Mominul, who hit six fours off 87 balls, when Duminy struck in his next over, trapping the left-hander leg-before for 30.
Mahmudullah was given out leg-before on 18 off Steyn but the decision was overturned on review as a replay showed a big inside edge.
You’ve reached your limit of {{free_limit}} free articles this month.
Subscribe now for unlimited access.
Already subscribed? Log in
Subscribe to read the full story →
Smart Quarterly
₹900
3 Months
₹300/Month
Smart Essential
₹2,700
1 Year
₹225/Month
Super Saver
₹3,900
2 Years
₹162/Month
Renews automatically, cancel anytime
Here’s what’s included in our digital subscription plans
Exclusive premium stories online
Over 30 premium stories daily, handpicked by our editors


Complimentary Access to The New York Times
News, Games, Cooking, Audio, Wirecutter & The Athletic
Business Standard Epaper
Digital replica of our daily newspaper — with options to read, save, and share


Curated Newsletters
Insights on markets, finance, politics, tech, and more delivered to your inbox
Market Analysis & Investment Insights
In-depth market analysis & insights with access to The Smart Investor


Archives
Repository of articles and publications dating back to 1997
Ad-free Reading
Uninterrupted reading experience with no advertisements


Seamless Access Across All Devices
Access Business Standard across devices — mobile, tablet, or PC, via web or app
