However, any hopes of achieving a positive result in what was already a weather-affected match, was ended by more rain on day five which prevented any play before lunch.
When Bangladesh returned to the crease in the afternoon, they lost their remaining seven first innings wickets inside 36 overs, slipping from 111-3 overnight to 256 all out.
Imrul Kayes took his score from 59 to 72, before being stumped by Wriddhiman Saha off the bowling of Harbhajan Singh.
One other bright note for Bangladesh came from the bat of debutant wicketkeeper Litton Das who hit a quick fire 44 from 45 deliveries before falling to Ashwin.
Spinner Ashwin added three more wickets on Sunday to post figures of 5-87 from his 25 overs ??? which was his first five-wicket collection outside of India.
The hosts, who fell seven runs short of saving the follow-on, were asked to bat again with a possible 30 overs available to the Indian bowlers.
But after Bangladesh openers, Tamin Iqbal and Imrul Hayes, had safely guided the team to 23 without loss, the visitors decided not to enforce another 15 optional overs and the game was drawn.
Rain had prevented any play on day two, while days one and four were also restricted at the Khan Shaheb Osman Ali Stadium.
Opener Shikhar Dhawan was made man of the match for his 173 from 195 balls, as India posted 462-6 declared.
The two teams are scheduled to play three ODIs in Dhaka on June 18, 21 and 24.