With no chance of a result in the weather-battered match, officials called off play at noon (0600 GMT).
Rain had washed out the entire fourth day's play and some 50 overs were lost on the second and third days at the Zahur Ahmed Chowdhury Stadium.
South Africa, who scored 248 in the first innings, bowled out Bangladesh for 326 before making 61-0 in their second knock.
Bangladesh's first innings total was their highest ever against the Proteas, the top-ranked Test team.
Stiaan van Zyl was unbeaten on 33 and Dean Elgar was 28 not out for South Africa.
The second and final Test will be played in Dhaka from July 30.