The West Indies dismissed the hosts for 287 and needing just 26 runs to win the match.
The chase was completed by openers Chris Gayle (20) and Kieran Powell (9) well before lunch on the fifth and final day.
Marlon was named man-of-the match for his career-best double century and Shivnarine Chanderpaul was man-of-the-series for his unbeaten 150 in the second test on top of the 203 not out in the first test last week in Dhaka. The West Indies won the first test by 77 runs.
Allrounder Shakib Al Hasan took four wickets in the second test becoming the highest test wicket taker for Bangladesh, overtaking spinner Mohammad Rafique. Hasan has now 102 test wickets. He also missed making a century by just three runs in the second test. Mohammad Nasir scored 94 runs before being bowled by Best.