Ben Stokes helped England complete a thrilling 22-run win over Bangladesh in the first Test on Monday.
Day five resumed with the hosts needing 33 runs, and the tourists two wickets, in Chittagong.
Stokes delivered for England, removing Taijul Islam (16) and Shafiul Islam (0) in the space of three balls early in the day.
It left Sabbir Rahman stranded unbeaten on 64 as Bangladesh were bowled out for 263 in their chase of 286.
Stokes (2-20) was the hero, having also taken four first-inning wickets and made 85 in the second.
England needed a review to get rid of Taijul Islam LBW, before Shafiul was trapped in front two balls later, and given out despite a review of his own.
The tourists lead the two-Test series 1-0 with the second match beginning on Friday.