Pakistan were poised to win the match comfortably after openers Abid Ali and Abdullah Shafique scored half-centuries to build a solid foundation.
Bangladesh's spin magician Taijul Islam trapped Abid Ali lbw on 91 runs while Abdullah Shafique was dismissed the same way by Mehidy Hasan on 73.
However, stable partnership between Azhar Ali, who scored 24 and skipper Babar Azam, who made 13, ensured Pakistan's win.
At stumps on Day 4, Pakistan finished at 109-0, needing only 93 runs on the final day.
Ali was 56 not out at the end of play, adding to his first-innings score of 133, while debutant Shafique finished unbeaten on 53.
"In Test cricket, a fourth day is very crucial," Shafique had said.
"Our plan today was to restrict them for the lowest total possible so that we can easily chase it down. This is what we were trying and we were successful."
Fast bowler Shaheen Afridi earlier claimed 5-32 to help Pakistan bowl out Bangladesh for 157 runs in their second innings despite wicketkeeper-batsman Liton Das scoring a fighting fifty.
Resuming on 39-4, Bangladesh looked down and out once they lost Mushfiqur Rahim in the first over of the day, before Liton, who struck a hundred in the first innings, hauled the hosts back to respectability with 59 off 89 balls.
Afridi trapped Liton leg-before and then had Abu Jayed out for a duck to complete his fourth five-wicket Test haul.
Off-spinner Sajid Khan, who finished with 3-33, wrapped things up by dispatching Taijul Islam.
The hosts had got off to a positive start with Mushfiqur hitting a boundary off the first ball of the morning, bowled by Hasan Ali.
But two balls later he left a delivery that darted back in and departed for 16 after Hasan uprooted his off-stump.
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