The big-hitting left-hander clubbed seven fours and eight sixes before ballooning to cover for 109 off 56 balls, while he shared a 168-run stand with Quinton de Kock, who contributed 63 off 38 deliveries.
The partnership was the highest in the tournament's history, eclipsing the 166-run union between Sri Lanka's Kumar Sangakkara and Mahela Jayawardene in 2010, as South Africa posted 205 for five.
Bangladesh succumbed to 101 all out in 16.3 overs in reply, with express fast bowler Anrich Nortje collecting four for 10 while left-arm wrist spinner Tabraiz Shamsi finished with three for 20.
South Africa were denied the chance to close out a victory against neighbours Zimbabwe in their tournament opener by the rain at Hobart on Monday, but they bounced back in style and next face India at Perth on Sunday.
On the same day, Bangladesh, who beat the Netherlands earlier this week, take on Zimbabwe at Brisbane.