The tourists posted 408 in their first innings in response to New Zealand's 553 for seven declared before the last wicket fell midway through the final session of the day.
In the five overs before stumps when bad light forced an early close, New Zealand had time to lose BJ Watling, run out for one by a diving throw by Bangladeshi captain Shakib Al Hasan.
They ended the day on nine for one in their second innings, an overall lead of 154 runs.
Bangladesh earlier went past the follow-on mark of 354 principally due to a defiant stand of 145 for the seventh wicket between 23-year-old Mahmudullah, who made 115, and Al Hasan, who scored 87.
These two had joined forces with Bangladesh in strife at 196 for six but they kept the home side wicketless for the entire second session after losing five wickets before lunch.
Al Hasan departed after tea in controversial circumstances when television replays indicated that an edge off Chris Martin did not carry to wicketkeeper Brendon McCullum, who claimed the catch.
There was nothing untoward about Mahmudullah's dismissal though as he was trapped leg before wicket by Dan Vettori after 192 minutes and 177 balls.
Vettori was the best of the bowlers with three for 88 while Martin had three for 116.
