The 22-year-old anchored the innings to notch up his maiden Test century as the hosts, replying to New Zealand's 469, cruised to 380-7 by stumps at the Zahur Ahmed Chowdhury stadium.
Skipper Mushfiqur Rahim (67) added 92 for the fifth wicket with Mominul to lift Bangladesh to 301-4, before both batsmen fell in the space of six deliveries after tea.
Debutant left-arm seamer Corey Anderson, who had conceded just 10 runs in his first 10 overs, was rewarded for his accuracy when he trapped Mominul leg-before.
Nasir Hossain (46) and Sohag Gazi (28 not out) frustrated the Black Caps further with a 70-run stand for the seventh wicket on a slow pitch that offered no assistance to the bowlers.
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The hosts, who have lost eight and drawn one of their nine Tests against New Zealand, trail by 89 runs with three wickets in hand.
Anderson was the lone New Zealand bowler to emerge with credit in unhelpful conditions, returning with figures of 2-23 from 14 overs.
Mominul, nicknamed Sourav by team-mates for a batting style that reminds them of former India captain Sourav Ganguly, hit 27 boundaries.
He fell just 19 runs short of becoming only the second Bangladesh batsmen after Rahim to score a Test double-century. The skipper Rahim hit 200 against Sri Lanka in Galle in March this year.