Ashwin was the pick of the bowlers with 2-30 while Harbhajan, back to the international cricket after two years, had figures of 1-23 as India pushed for a result after the rain kept coming at regular intervals.
Because of the rain, lunch was taken 10 minutes before the scheduled time.
Opener Imrul Kayes was not out on 59 with Shakib Al Hasan yet to score as Bangladesh sought to avoid the follow-on and grind out a draw.
Mominul Haque made 30 while opener Tamim Iqbal scored 19.
The in-form Tamim began fluently after India declared its innings ahead of the start of the day's game.
His aggressive shot-making however led his downfall as he was lured by a flighted delivery from Ashwin and wicketkeeper Wriddhiman Saha made an efficient stumping.
Ashwin later had Mushfiqur (2) trapped leg-before with a classic off-spin just before the rain arrived.
Harbhajan dismissed Mominul Haque who played a rash shot to give away his wicket after adding 81 runs with Kayes.
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