Pacer Romario Shepherd (3-36) became the second West Indies bowler to take a Twenty20 hat-trick after teammate Jason Holder.
West Indies reached 5-152 in 16.5 overs after Bangladesh were bowled out for 151 in 20 overs in the third Twenty20 match. Bangladesh opener Tanzid Hasan scored 89 off 62 balls, smashing nine fours and four sixes.
Auguste was out for a 25-ball 50, including five sixes. Chase was also out for 50, off 29 balls. The pair combined for 91 runs for the fourth wicket.
Leg-spinner Rishad Hossain dismissed both of them in three balls, leaving the visitors eight runs from victory.
Rovman Powell and Gudakesh Motie wrapped up the victory for West Indies, who had lost the preceding ODI series 2-1.
Shepherd's three wickets came across two overs. He dismissed Nurul Hasan for one in the 17th over, then returned to bowl the last over and dismissed Tanzid and Shoriful in the first two balls.
Bangladesh captain Liton Das had won the toss and opted to bat first. The hosts appeared headed for a big total when Saif Hassan (23) and Tanzid added 63 for the third wicket.
Holder (2-32) got the better of Hassan and sparked a collapse that saw Bangladesh lose the last seven wickets for 33 runs.
West Indies won the second Twenty20 match by 14 runs and the opener by 16 runs.
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