Needing 19 off the final over, Carlos Brathwaite hit Ben Stokes for four consecutive sixes as West Indies finished on 161 for six, chasing a victory target of 156.
West Indies won by four wickets and the victory was achieved with two balls to spare.
Brathwaite finished on 34 not out from 10 balls, while Marlon Samuels top scored with 85 from 69 balls. Samuels hit nine fours and two sixes to steady the early slump, while Brathwaite had four sixes and a four.
Samuels and Brathwaite saved West Indies after the Caribbean side struggled throughout after losing explosing opener Chris Gayle for four, Johnson Charles for one and Lendl Simmons without scoring. At that stage, West Indies were struggling on 11 for three off 2.3 overs.
It was 86 for four off 13 overs when Dwayne Bravo was dismissed for 25 off 27 balls following an important 75-run partnership with Samuels. That partnership took the score from 11 for three.
David Willey was the best bowler for England with three for 20 from his four allotted overs.
Earlier, England sent in to bat made 155 for nine off their allotted 20 overs after they had a disastrous start.
England lost their first wicket without a run on the board when right-arm leg spinner Samuel Badree bowled Jason Roy with the second ball of the match.
They were struggling at two for eight after 1.5 overs, then three for 23 after 4.4 overs and 84 for four after 11.2 overs.
Bravo, a medium pacer, and Brathwaite picked up three wickets each.
Brathwaite finished with three for 23 from his four allotted overs, while Bravo had figures of three for 37 off four.
Badree finished with two for 16 from four, while Andre Russell had one for 21 off four.
Joe Root top scored for England with 54 from 36 balls.
Both sides were attempting to become the first country to win two World T20 titles and the West Indies achieved history.
This was the second meetings between the teams in this tournament.
In the first game, Chris Gayle tormented the England bowlers by smashing an unbeaten century from only 48 balls to guide West Indies to a six-wicket win in Mumbai.
The victory means that West Indies have secured the double after West Indies Women chased down the target of 149 in 19.3 overs with eight wickets in hand to beat Australia earlier.
Earlier this year West Indies won the Under-19 World Cup in India. They beat India, the pre-tournament favourites, by five wickets in Mirpur to lift their maiden Under-19 World Cup title.