Rain brought an early end to proceedings in Christchurch, with the Black Caps being reduced to 260-7 in their first innings before play was halted.
The hosts had earlier found themselves on a healthier 252-4 before Al Hasan struck to halt their momentum, giving Bangladesh further hope when play resumes on Sunday evening.
New Zealand saw two men fall to Kamrul Islam Rabbi within the space of three balls, but Tom Latham and Ross Taylor put together a 106-run third-wicket partnership, only to then fall in quick succession for 68 and 77 runs respectively.
Al Hasan struck three times late on after seeing Henry Nicholls (56*) and Mitchell Santner (29) also pile on the runs, with Jeet Raval (16), Kane Williamson (2) and BJ Watling (1) failing to cause any damage.
Colin de Grandhomme also went for a duck, as New Zealand's solid start soon faded to leave the hosts trailing by 29 runs with three wickets still remaining from their first innings.