The 31-year-old Barbadian followed his maiden five-wicket haul in the first Test win in Dhaka with 6-40 to dismiss the home team for 287 in the second innings on the fifth and final morning at Sheikh Abu Naser Stadium.
The tourists knocked off the required 27 runs in just 4.4 overs to win their second successive series in Bangladesh. They also won the two-Test series 1-0 last year.
Chris Gayle finished with 20 not out while fellow opener Kieran Powell (nine not out) sealed the win with two consecutive boundaries off Naeem Islam.
Bangladesh avoided an innings defeat thanks to a fighting 94 by Nasir Hossain but lost their last four wickets with the addition of 61 runs after resuming at 226-6.
It is Bangladesh’s 65th defeat in 75 Tests since gaining Test status 12 years ago — a dismal record which constantly raises questions over their right to play the five-day version of the game.
Their losses include 34 defeats by an innings.
West Indian captain Darren Sammy described the series win as welcoming.
“It is a step in the right direction and the team played well, it was remarkable to see the way we played,” said Sammy, who also led the team to 2-0 Test win over New Zealand followed by the World Twenty20 triumph last month.
“Best has put us in good positions and it is good to see everyone putting in a lot of hard work,” he added.