Rabada 'a superstar' as South Africa crush Bangladesh
Rabada took 6-46 as Bangladesh went from 283-7 overnight to 307 all out, leaving the visitors a victory target of 106.
Rabada finished with 9-72 in the match, having become the fastest bowler to reach 300 Test wickets during the first innings.
"It's special," stand-in captain Aiden Markram said of a victory that was sealed before lunch on day four and gave South Africa a 1-0 lead in the two-match series.
They move up to fourth in the World Test Championship standings.
"A pretty young group, or slightly inexperienced group... to come to the subcontinent and get a win is great for us," Markram added.
Left-arm spinner Taijul Islam took all the wickets to fall in his 3-43, but he failed to slow South Africa's chase as they reached 106-3.
Tony de Zorzi top-scored with a quickfire 41 off 52 balls, hitting seven fours.
Fellow opener Markram was first to go, bowled by Taijul for 20.
De Zorzi departed in the 16th over trying to play a big shot over long-on, caught by Hasan Mahmud.
David Bedingham was third out caught behind by wicketkeeper Litton Das.
Tristan Stubbs, unbeaten on 30, ensured there were no further alarms.
"He has been a superstar for us for years," Markram added of Rabada.
"And to see all those stats pop up about him... it really proves that he's one in a million, and we're really lucky to have him on our team."'Disappointing'
Bangladesh lost their last three wickets for just 24 runs inside half an hour, all-rounder Mehidy Hasan Miraz top-scoring with 97.
"Missing out on the century is disappointing," said Mehidy.
"I would have been happy if I had executed the plan better. I was only focusing on guiding the team to a strong position."
Nayeem Hasan departed lbw without adding to his overnight 16 on the third ball of the day when a rampant Rabada inswinger kept low.
Wiaan Mulder removed Taijul for seven in the fourth over via a thick edge to Stubbs at third slip.
Mehidy fell three runs short of his second Test century in the next over when he edged Rabada to Mulder at slip.
South Africa scored 308 in their first innings, powered by Kyle Verreynne's second Test century, in reply to Bangladesh's 106 all out on the first day.
Bangladesh have still never won a Test against South Africa.
The second and final match of the series begins on October 29 in Chattogram.