The 31-year-old paceman took three wickets in the post-lunch session and added another in the final session, finishing with 5-24 to rout Bangladesh for a paltry 167 after they were set a 245-run target.
Best dismissed Shahriar Nafees (23), Shakib Al Hasan (two) and Mushfiqur Rahim (16) in quick succession and then had Bangladesh's top scorer, Mohammad Mahmudullah (29), to give his team a 1-0 lead in the two-Test series.
The second and final Test will be played in Khulna from Wednesday.
"Bangladesh played really well, but credit must go to our bowlers and batsmen. It turned out to be a very good wicket after 1000 runs in the first two innings, but there was something in the final stages of the match," said Darren Sammy, the West Indies captain.
Sammy's Bangladesh counterpart, meanwhile, bemoaned a lack of partnerships in his team's second innings.
"In this Test, we competed well ... but lack of partnerships in the second [innings] let us down," said Rahim.
Bangladesh were put on course for a win by off-spinner Gazi whose 6-74 are the best figures by a home bowler on debut.
Gazi had Sammy in the third over of the day after the tourists resumed at 244-6 to raise home hopes. But Bangladesh could not finish the job with the bat.
* Agence France-Presse
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