New Zealand were reduced to 7-113 at stumps on day four, chasing an unlikely victory target of 332 on a turning pitch.
Daryl Mitchell batted patiently to end the day on 44 on Friday, and looks to be New Zealand's only realistic hope with Ish Sodhi at the other end on seven. Opener Devon Conway had, earlier, been the only other batter to put up resistance with 22.
"It's not a bad wicket. I suppose their bowlers have bowled very disciplined and put the ball in good areas and asked us good questions for long periods of time," New Zealand spinner Ajaz Patel said.
"A lot of batting to do tomorrow. We've got Dazzler (Mitchell) there who's batted quite nicely today. And we know how capable Ish is with the bat as well.
"I think we pride ourselves to fight all the way through. So, it'll be an interesting day tomorrow."
Daryl Mitchell (left) finishes the day in Sylhet as New Zealand's last hope on 44 not out. (AP PHOTO)
But Bangladesh will take some stopping as they scent a big win.
"We need this victory badly because when you will beat a big team like New Zealand, it will give you the confidence," the potential matchwinner Taijul said.
"We won't say we've won the game until we grab the last wicket."
Pacer Shoriful Islam extended Tom Latham's lean patch with a six-ball duck in the first over, and Kane Williamson failed to build on his first-innings century on an increasingly deteriorating wicket.
Taijul troubled Williamson by varying the pace and finally trapped him lbw for 11, with the master batter reviewing in vain.
Henry Nicholls was the next to go, top-edging a slog-sweep off Mehidy to Nayeem Hasan at short fine leg for two.
Conway perfectly swept the spinners to keep the scoreboard moving before Taijul had him caught by Shahadat Hossain at short leg.
With New Zealand at 4-46, the onus was on Mitchell but he found little support from the other end.
Wicketkeeper-batter Tom Blundell (six) was stifled by Taijul before edging a turning delivery behind, and Nayeem pegged back the visitors further with a successful review to remove Glenn Phillips.
When Taijul got the better of Kyle Jamieson for seven, Bangladesh looked like they could wrap up the win on day four, but Mitchell and Sodhi survived a tricky 30 minutes before the bad light forced stumps.
Earlier, Bangladesh were bowled out for 338 in their second innings. Najmul Hossain Shanto added just one run to his overnight score and was out for 105 as Bangladesh lost a succession of wickets.
Mehidy Hasan scored 50 not out to help the side take the lead past 300.
Mushfiqur Rahim continued and raised his 27th Test half-century off 79 balls, with an outside-edge two against Southee but Patel trapped him in front for 67.
