Despite Zakir Hasan's century, India need just four wickets on the final day to beat Bangladesh in Chattogram.
Axar Patel struck three times to leave hosts Bangladesh on 272-6 at the end of day four of the first Test.
Chasing a huge target of 513 at Zahur Ahmed Chowdhury Stadium after tons from India's Shubman Gill and Cheteshwar Pujara, Bangladesh reached the end of day three with 42 runs on the board for no loss.
That stubborn resilience continued on Saturday as they made it to 124-0, before Najmul Hossain Shanto (67) was finally removed by Umesh Yadav early in the second session.
Yasir Ali was swiftly dismissed after facing just 12 deliveries, Axar picking up the first of his three wickets.
Zakir watched another of his partners fall when Litton Das (19) sent a Kuldeep Yadav googly into the waiting hands of Umesh at long-on.
The 24-year-old reached his debut hundred with a smart sweep for four, but an inside edge to Virat Kohli from Ravichandran Ashwin's full ball saw Zakir's stand come to an end.
Bangladesh's slim chances of victory were further dashed by a brilliant over from Axar.
The left-arm spinner first took out Mushfiqur Rahim's off stump before luring Nurul Hasan out of his crease, allowing Rishabh Pant to smash off the bails.
Axar finished day four with figures of 3-50 off 27 overs, with Bangladesh needing 241 runs from the final day for an unlikely victory, with captain Shakib Al Hasan and Mehidy Hasan Miraz at the crease.
Axar the pick of the bunch
Axar was the only India bowler to take more than one wicket, including those two in a superb 88th over of the innings.
He went for just 1.85 runs an over and is sure to play an important role on day five.
Zakir stars for hosts on debut
In his maiden Test for Bangladesh, Zakir was the man who kept the very slim possibility of a win for the hosts alive.
Bangladesh frustrated India in the first session, with Zakir's steady hands guiding them to 119-0 by the end of the first session.
While his team's innings eventually started to crumble around him, Zakir dug in to reach a memorable 100, which included 14 boundaries (13 fours, one six).