Best helped West Indies skittle Bangladesh for 287 runs in their second innings on the fifth day, setting up a target of 28 runs to win for the tourists, who won the first test by 77 runs.
The West Indies reached 30-0 in just 4.4 over complete a 2-0 sweep in the series.
Kieran Powell smashed Naeem Islam for two successive fours to bring up the win while Chris Gayle was unbeaten on 20 at the other end.
Bangladesh resumed play at 226-6 and lost a wicket on the fourth ball of the morning when Mahmudullah was caught by Denesh Ramdin off Veerasammy Permaul for two.
Best added three wickets in the morning to his three overnigh, giving him a five-wicket haul in successive tests.
He struck with his first ball of the day, bowling Sohag Gazi for seven before the paceman uprooted the leg stump of Nasir Hossain with the batsman on 94.
Nasir, who began the day on 64, has now missed his maiden century in second successive test after he was out for 96 in the first innings of first test in Dhaka.
Best removed the final wicket of Rubel Hossain for 14 a ball after the batsman was dropped by Fidel Edwards off Permaul.
Marlon Samuels was named man of the match for his 260 runs in the first innings. Shivnarine Chanderpaul, who scored a double century and a century in two tests, was named man of the series.
The five-match ODI series between the teams begins on Nov. 30.