Zakir Hasan (40), Najmul Hossain Shanto (38) and Mominul Haq (34) chipped in with useful cameos as Bangladesh chased down the 185-run victory target with more than a session left in the game.
Chasing 185 to win, the visitors inched their way to the target for the loss of four wickets, 25 minutes before tea on the fifth and final day to complete a 2-0 series sweep.
At the start of play on the fifth and final day of the match, Bangladesh resumed their innings at 42-0 in pursuit of their 185-run target. Pakistan got the first breakthrough after 16 runs were added when Mir Hamza bowled Hasan with a delivery that moved away.
Fellow pace bowler Khurram Shahzad got in on the act when Shadman Islam miscued a drive, chipping to Shan at mid-off.
However, a lack of further breakthroughs and the shallow target meant that Bangladesh tightened their grip on the match and the series, reaching the 100-run mark in the 29th over.
Najmul Hossain Shanto on 33 and Mominul Haque, 20, dug in to stop any further inroads with the captain bringing up their fifty partnership off the last ball before lunch.
Shanto fell after the lunch break and Mominul threw away his wicket but veterans Mushfiqur Rahim, who made 22 not out, and Shakib Al Hasan guided Bangladesh home without any drama.
Shakib, who finished 21 not out, sealed the victory by hitting spinner Abrar Ahmed for four through the covers.
Bangladesh leads the two-match series 1-0 after a surprise win in Rawalpindi last month, their first in 14 tests against Pakistan.
