Rain and bad light had also forced early finishes to the second and third days, eating up time and making a result unlikely.
South Africa made a slow start to their second innings after conceding a first innings lead of 78 runs.
Opening batsmen Stiaan van Zyle was on 33 with Dean Elgar on 28 with South Africa 61 without loss.
The Shere Bangla National Stadium in Mirpur will host the second and final Test from next Thursday.
This is historic! Women stars to make debut on FIFA video game cover
Why Saha is the right choice to fill Dhoni's big shoes
Best Pictures from the Tour de France - Part 1
Special Olympics athletes made to sleep on the floor!