Bangladesh, who needed to bat out their second innings after being set an improbable victory target of 467, were 271-3 when stumps were drawn on the fifth day at the Zahur Ahmed Chowdhury stadium.
Mominul, a 22-year-old left-hander, held the innings together till both sides accepted the draw as soon as he reached 100, his third Test hundred, the post-tea session.
Shakib Al Hasan returned unbeaten on 43 as he and Mominul added 120 for the fourth wicket after Bangladesh were 151-3 before tea.
It was only the second draw in 16 Tests between the two countries, with Sri Lanka having won the other 14 matches, eight of them by an innings margin.
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Bangladesh, who languish at the bottom of the Test rankings at number 10, have lost 68 of their 83 Tests since gaining full status in 2000, winning just four.
Bangladesh captain Mushfiqur Rahim said he was delighted with the draw and hoped the momentum gained by his side will spill over to the limited-overs contests.
"We may have lost the series but I am very happy at the way the team applied themselves. I hope we can carry this momentum into the one-day and Twenty20 series.