Bangladesh took two important New Zealand wickets before lunch on Tuesday to raise hopes of what would be their first-ever win against a major Test-playing nation.
New Zealand battled hard to ensure an exciting finish to the first Test at the Divisional Stadium in Chittagong, but will need to improve on their previous best run chase away from home to pull off a victory.
Chasing a tough 317-run target, they lost Aaron Redmond (79) and then Ross Taylor (nine) to reach 212-4 at lunch on the fifth and final day, needing 105 more runs for a win.
Skipper Daniel Vettori held steady for the tourists with a fighting 36 and Brendon McCullum was unbeaten on one at the break after New Zealand resumed at 145-2.
If New Zealand win they will better their previous best run chase in a Test away from home when they made 164-4 against Australia at Perth in 1985.
They opened the final day with just that intention as Redmond and Vettori saw off the first hour without loss until Bangladesh hit back through first innings hero Shakib Al Hasan.
The left-arm spinner, who set a new Bangladesh best innings record of 7-36, had one turn sharp across Redmond's bat and the ensuing edge was smartly held low by Zunaed Siddiqui at slip.
Redmond hit 10 boundaries during his 237-ball knock and added 40 runs for the third wicket with Vettori.
Bangladesh skipper Mohammad Ashraful took the second new ball after 83 overs and was instantly rewarded by Masharfe Mortaza who dismissed Ross Taylor, caught at cover by substitute fielder Mahbubul Alam.
Bangladesh's only win in their previous 53 Tests came against Zimbabwe here three years ago.