The victory gave Towhid Hridoy’s side the series after rain washed out the first two matches.
Bangladesh’s spinners claimed seven wickets between them, with Rishad taking four for 45 and Alis three for 35.
After winning the toss and opting to bowl, Bangladesh struck in the sixth over through Musfik Hasan, who removed Rivaldo Moonsamy.
Alis then took charge, dismissing Zubayr Hamza lbw before trapping Sinethemba Keshile in front and bowling Matthew de Villiers with successive deliveries.
South Africa A were reduced to 53 for four before Marcus Ackerman attempted to rebuild the innings. He scored a cautious 54 off 67 balls, but Rishad eventually removed him lbw. Imran Manack added 43 off 53 balls, but South Africa A were bowled out for 182.
Bangladesh made a flying start in the chase, with Saif Hasan and Naim Sheikh putting on 92 for the opening wicket.
Saif raced to his half-century from 40 balls and went on to score 61 off 53 before being stumped off Simond.
Naim reached his fifty from 70 balls but departed for 60 off 77, while Parvez Hossain Emon was dismissed for 17.
Nurul Hasan Sohan then scored an unbeaten 16 to help Bangladesh complete the chase comfortably.
