Evin Lewis smashed a 48-ball century and Johnson Charles scored 79 off 33 deliveries in West Indies' 245-6 in the series opener in Lauderhill, Florida.
KL Rahul led India's reply with an unbeaten 110 - his 46-ball hundred was the joint second fastest in T20s.
But Dwayne Bravo had MS Dhoni caught at short third man off the final ball of the match to seal a remarkable victory.
Lewis, the 24-year-old left-handed opener playing only his second international match, took 32 off a Stuart Binny over, the joint second most expensive over in T20s.
West Indies hit 21 sixes - a record in a T20 international innings - while their total was the third highest in history. India's response sits fourth on the list.
The final game of the series takes place at the same venue on Sunday. India won the four-Test series 2-0.
West Indies held on to beat India by one run in a thrilling Twenty20 that saw a record 489 runs and 32 sixes.