West Indies declared their second innings on 269 for four after Shivnarine Chanderpaul (101 not out) had completed his 30th Test century.
Jermaine Blackwood (66) had helped put on a 169 run fourth wicket partnership as West Indies set a near impossible target of 488.
The more realistic challenge for Bangladesh was to bat for as long as possible and hope the unpredictable weather in Saint Lucia at this time of year might help them.
After falling to 48-2, the third wicket pairing of Mominul Haque (56) and Tamim Iqbal (64) dug in and put on a 110 run parternship.
But when Tamim went, top-edging Sulieman Benn to Shannon Gabriel, Bangladesh quickly fell apart.
The tourists added 34 runs for their final seven wickets with Benn's left-arm spin bringing him figures of 5-72 while seamer Jerome Taylor chipped in with 3-39.
It was far too easy for West Indies, as it was throughout a series which, while it may have boosted the confidence of some of the newer faces in the Caribbean, did little for Bangladesh as they struggle to establish themselves in Test cricket.