The match was scheduled to start at 9:30am at Zahur Ahmed Chowdhury Stadium on Friday but was abandoned due to overnight rain soaking the port city.
The players, too, did not come to the stadium from the hotel.
It is for the first time that a Test match between these two sides has stretched to the fifth day.
The Proteas are trailing the Tigers by 17 runs.
With the fourth day ending without a ball being bowled, the tourists’ chance of winning the match has become even slimmer.
South Africa were 61 for no loss on Thursday, when bad light forced the players off the field. The play was called off later because of the rain.
Openers Stiaan van Zyl (33) and Dean Elgar (28) had cut the deficit to 17 runs.
Bangladesh had started the third day at 179 for 4 and were bowled out for 326, their highest score against South Africa, in the second session.
The Tigers restricted South Africa to a mere 248 in the first innings, thanks to some magnificent bowling by Bangladeshi bowlers.
Debutant Mustafizur Rahman returned 4 for 37, which included three wickets in four balls, while legspinner Jubair Hossain got 3 for 53.
Plays of the previous two days were also cut short by the rain.