Du Plessis helped the visitors recover from a shaky start with an unbeaten 79 off 61 balls, his sixth half-century in Twenty20 internationals, which included eight fours.
Off-spinner JP Duminy (2-11), medium pacer David Wiese (2-12) and fast bowler Kagiso Rabada (2-28) then put in a combined effort to dismiss Bangladesh for 96 runs on a slow wicket at the Sher-e-Bangla National Stadium.
Shakib Al Hasan was the only batsman to offer some kind of resistance with 26 off 30 balls while debutante Litton Das made 22 off 33 balls.
The game decisively swung the Proteas' way when Duminy dismissed Rahim and Sabbir Rahman (4) in his successive overs and left-arm spinner Aaron Phangiso removed Nasir Hossain for one as Bangladesh collapsed from 2-50 to 5-57.
Earlier, opting to bat first, South Africa struggled to keep the scoreboard rolling.
Bangladesh started with spinners from both ends and struck almost immediately with left-arm spinner Arafat Sunny dismissing AB de Villiers in the very first over for two.
Quinton de Kock tried to break the shackle with a six off Nasir over cover but soon became a catch of Das off the same bowler at point for 12.
