Afif Hossain made 72 to lead a recovery by Bangladesh after Rabada, Lungi Ngidi and Wayne Parnell ripped through their top order.
Bangladesh, who won the first match by 38 runs in Centurion on Friday, crashed to 34 for five after visiting captain Tamim Iqbal won the toss and decided to bat.
Four of the first five wickets fell to deliveries which bounced sharply off the pitch.
South Africa were hampered by an injury to Parnell, who left the field with a hamstring injury five balls into his third over.
South Africa had picked only five bowlers. Rabada had already bowled seven of his ten overs in his opening spell of three for 25 and Bavuma had to rely on his slow medium-pacers and the spin of Tabraiz Shamsi and Keshav Maharaj for much of the innings.
With the pace threat negated, the left-handed Afif batted impressively as he shared partnerships of 60 with Mahmudullah (25) and 86 with Mehidy Hasan (38). Afif, 22, hit nine fours in a 107-ball innings before falling to Rabada late in the innings.
AFP