They surpassed the previous record of 263-5, set by Royal Challengers Bangalore in 2013
Bangladesh were sent in to bat after the West Indies won the toss at Lauderhill, Florida on Saturday.
Half-centuries from Tamim Iqbal (74) and captain Shakib Al Hasan (60) helped the Tigers to a score of 171/5, before the spinners restricted the West Indies to 159/9.
Nazmul Islam was the best of Bangladesh’s bowling attack, taking three wickets for 28, while Mustafizur Rahman picked up three for 50 and Shakib scalped two for 19.
ADS Fletcher (43) and Rovman Powell (43) had built a 58-run partnership that threatened the Tigers, but the West Indies faltered after Nazmul took Fletcher’s wicket in the 15th over.
Tamim Iqbal was declared Man of the Match.
The deciding match in the series will also be played in Lauderhill on Monday Bangladesh time.