Bairstow's unbeaten 108 off 48 balls, with eight fours and nine sixes, see Punjab secure the win with eight balls to spare
Left-hander Mominul equalled Tamim Iqbal's haul of nine centuries in the longest format on Monday.
He stitched together a 200-run stand with Mushfiqur, who hit his seventh Test century, to take the Tigers' first innings lead past 100 on Day 3.
Bangladesh are currently 370-3.