The hosts reached 269-3 in their second innings at stumps on the fourth day, wiping out New Zealand's first innings advantage of 155 for an overall lead of 114 runs.
Mominul led Bangladesh's batting charge with 126 not out, making him only the second Bangladeshi to score back-to-back Test centuries after Tamim.
Mominul, who scored 181 in the first innings of the drawn first Test in Chittagong, batted solidly throughout the day on Thursday, showing no sign of nerves until he got stuck on 99 for 11 balls.
Former captain Shakib Al Hasan was accompanying Mominul with 32 not out when stumps were drawn on the penultimate day of play.
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Tamim, who achieved the same feat at Lord's and Old Trafford against England in 2010, played a patient knock of 70 off 218 balls before Ross Taylor spectacularly caught him one-handed at slip off Kane Williamson.
Mominul and Tamim added 157 runs, a record for Bangladesh for the third wicket, after Wagner struck twice in the first session to leave the home side struggling on 55-2.
Anamul's 22, his highest score in the series, included four boundaries, while Ayub fell for just nine as Taylor took a catch at slip.
Earlier New Zealand had resumed on 419-8 but added just 18 for the last two wickets to be all out for 437.
Ish Sodhi was run out by Mominul for 58, before Abdur Razzak trapped Trent Boult in front to wrap up the innings seven overs into the morning session.