The 22-year-old was unbeaten on 172 as the hosts, replying to New Zealand's first innings total of 469, cruised to 272-4 by tea at the Zahur Ahmed Chowdhury stadium.
Skipper Mushfiqur Rahim returned undefeated on 49, having added 92 for the fifth wicket with Mominul that left the New Zealand bowlers in disarray.
Bangladesh, who have lost eight and drawn one of their nine Tests against New Zealand, trail by 197 runs with six wickets in hand on a slow pitch that has offered the bowlers no assistance.
Fellow newcomer Ish Sodhi went for 68 runs in 16 overs of leg-spin, while left-arm slow bowler Bruce Martin conceded 51 runs in 13 overs.
Mominul, nicknamed Sourav by team-mates for a batting style that reminds them of former India captain Sourav Ganguly, has so far hit 26 boundaries.
He is now just 28 runs away from joining his skipper as the only two Bangladeshi batsmen to score a Test double-century. Rahim hit 200 against Sri Lanka in Galle in March.
Mominul put on 126 for the third wicket with debutant Marshall Ayub, who defended dourly at the other end while making 25.
The pair, resuming at 103-2, took the score to 134 when Ayub was caught behind to hand Anderson his first Test wicket.
Former captain Shakib Al Hasan scored 19 in a fourth-wicket stand of 46 with Mominul when he was caught behind off Kane Williamson in the first over after lunch.
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