Bangladesh were dismissed for 307 in their second innings during the first session on Thursday, with Rabada grabbing his six-for and leaving South Africa needing 106 for victory.
The tourists completed the inevitable by reaching 3-106 in 22 overs just before lunch on day four.
Bangladesh were always chasing after being dismissed for 106 in their first innings. South Africa replied with 308, propped up by Kyle Verreynne's 114, which resulted in him being named player of the match.
"It's a really good performance. Incredibly tough for the team to put four days of good cricket together," South Africa stand-in captain Aiden Markram said.
"Bowlers were great there and then the batters helped us get a great lead. When you get a lead like that, it's tough for opposition to come back."
South Africa's last Test win in Asia was in 2014 against Sri Lanka in Galle.
Resuming the day at 7-283 and 81 runs ahead, Bangladesh failed to make any impact as Rabada and Wiaan Mulder sliced through the hosts with the second new ball.
"Rabada is a special bowler for us," Markram said.Â
"For Kyle to step in and play that sort of innings (114), that's really special. (The) emphasis is on whoever's playing should make it count."
Rabada struck in the first over of the day, trapping Nayeem Hasan lbw for his overnight score of 16 to complete his 15th five-wicket haul in Test cricket. He finished the match with nine wickets.
Bangladesh left-arm spinner Taijul Islam took eight wickets overall, after picking up all three South African wickets on Thursday.
"As a batting group we need to take responsibility against the new ball, and as a bowling group we need to show improvement," Bangladesh captain Najmul Hossain Shanto said.
"We need to put in a collective performance in the next Test match."
The second and final Test starts in Chattogram on Tuesday.
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