India declared their second innings at 100-6 when play began on the fifth and final day at 2:00 pm (0800 GMT), after four hours were lost due to damp ground conditions caused by heavy overnight rain.
India, looking for quick runs after resuming at their overnight score of 44-2, lost four wickets in 10 overs but still managed to add 56 runs.
Bangladesh need 5.9 runs an over to win, weather permitting. Dark clouds were gathering as the hosts began their second innings.
Bad weather had already taken a toll on the match with only 20 overs played on the second day and the entire third day's play washed out.
The second and final Test starts in Dhaka on Friday.