Tamim, who spent one season in Wellington, wasted no time getting on the front foot with a string of bold strokes against an out-of-sorts Boult.
He was strong through the point and cover area, eased left armer Boult to the wide mid on boundary and off drove medium pacer Colin de Grandhomme to the long off fence.
He had hit seven boundaries, and also survived a DRS referral from New Zealand after a loud appeal when de Grandhomme struck him on his front pad.
Replays suggested the ball passing over the leg stump.
Mominul, who scored two centuries against New Zealand in Bangladesh in 2013 and is averaging 51 from his 19 tests, was battling hard as Southee gave him a decent examination.
Southee had taken one for two from his 5.3 overs while Boult's opening three overs cost 26.
That raised the question of whether bowling into the buffeting breeze, as Southee did, was easier to retain balance than bowling with the wind at the back, as Boult had. The respective performances suggested Southee had the better end of the deal.
This is Bangladesh's third test at the Basin. The first two, in 2001 and 2008, ended in heavy innings' defeats.
New Zealand are chasing a third win in as many home tests this summer, having beaten Pakistan 2-0 before Christmas. They have never been beaten by Bangladesh at home in any form of cricket, the record standing at 19 matches, 19 wins.