Das had been rollicking along to reach 40 from 46 balls before slowing down as his 11th half-century neared.
Only once before in nine Tests in New Zealand have Bangladesh held a first-innings lead but on that occasion -- in Wellington four years ago -- the second innings did not go their way and they lost by seven wickets.
Neil Wagner and Kyle Jamieson have been the most likely New Zealand bowlers, with Wagner accounting for Mahmudul Hasan Joy for 78 and no-balled when he had Mominul caught behind.
Jamieson trapped Mominul in front with the second ball after lunch but the ball had pitched just outside leg stump, and a Mominul edge off Trent Boult landed just short of Ross Taylor.
Boult did have success with the wicket of Mushfiqur Rahim, bowled for 12, but had trouble trying to contain Das, who hit the left-armer to the fence on both sides of the wicket.
Mominul was equally harsh on Tim Southee in his first spell after lunch with back-to-back fours, one through covers and one to third man to underscore the freedom with which the batsmen were scoring.