Only 221 overs could be bowled in the entire match, with rain curtailing play on the second and third days before washing out the entire fourth day's play on Friday.
Steady rain washed out the entire morning session on Saturday, prompting the match officials to call off play with South Africa on 61 for no loss in their second innings.
Bangladesh are unlikely to mind the outcome considering they lost to South Africa, who are currently eight places above them in Test rankings, in their eight previous meetings.
Pitted against the world's top Test team, they had stunned in the preceding One-Day series and Bangladesh underlined their recent improvement in the longer format as well, bowling out the tourists for 248 and taking a 78-run first innings lead.
It was the kind of display that proved their recent rise in form, including home ODI series victories against Pakistan and India, were far from fluke.
They seem to have unearthed a pace talent in left-arm bowler Mustafizur Rahman, who announced his Test arrival claiming three wickets in four balls in South Africa's first innings.
The Shere Bangla National Stadium in Mirpur will host the second and final Test from Thursday.
Scoreboard:
South Africa 1st innings: 248 (T. Bavuma 54; M. Rahman 4-37)
Bangladesh 1st innings: 326 (Mahmudullah 67, T. Iqbal 57, L. Das 50)
South Africa 2nd innings (Overnight: 61-0)
S. van Zyl not out 33
D. Elgar not out 28
Extras 0
Total (for no loss, 21.1 overs) 61
Did not bat: H. Amla, F. du Plessis, T. Bavuma, Q. de Kock, J. Duminy, V. Philander, D. Steyn, S. Harmer, M. Morkel
Bowling: M. Rahim 5-0-21-0, T. Islam 2-0-4-0, Mahmudullah 1-1-0-0, S. Al Hasan 5-0-19-0, M. Shahid 6-1-12-0, J. Hossain 2.1-1-5-0.
Result: Draw