Pakistan, who won the first Test in Chittagong by an innings and 184 runs, achieved their 103-run victory target off just 20.5 overs in the last session of the fifth day's play at the Sher-e-Bangla stadium.
Abdur Rehman celebrates the dismissal of Mushfiqur Rahim during the second Test.
Opener Mohammad Hafeez smashed a 52-ball 47 and Azhar Ali contributed 34, before skipper Misbah-ul-Haq finished the match with a six over long-off against Elias Sunny.
Bangladesh, trailing by 132 runs after the first innings, had been comfortably placed at 212-5 in their second innings but collapsed to 234, with left-arm spinner Rehman taking three wickets and off-spinner Saeed Ajmal two. Rehman's four-wicket haul was his best in 12 Tests.
Nasir Hossain (79) and skipper Mushfiqur Rahim (53) defied the Pakistani attack for more than a session, adding 117 for the sixth wicket before falling in the space of nine runs.
Rehman removed both the well-set batsmen in four overs before accounting for Shahadat Hossain (one) in the afternoon session.
Nasir, a 20-year-old playing only his fourth Test, cracked 11 fours in his maiden half-century before being bowled by Rehman. Rahim hit seven boundaries in his ninth Test half-century.
Rahim reached his half-century when he swept Rehman for a four, but soon lost his patience, stepping out to attempt a lofted shot off the spinner to be caught by Ajmal at mid-off.
Rehman got another wicket in his next over when he dismissed Shahadat, superbly caught by Younis Khan at first slip, while Ajmal bowled Sunny and had Robiul Islam stumped to complete 50 Test wickets in the year.
Nasir and Rahim had applied themselves well in the morning session to keep the Pakistani attack at bay as the hosts added 86 runs to their overnight total of 114-5 after a delayed start due to fog.
