The opener struck an unbeaten 103 off 54 balls on the island he used to call home as England cruised to a pursuit of 183.
It was the opener's third T20 hundred for England - all three of which have come against West Indies in the past 12 months.
Warwickshire’s Jacob Bethell, another with Bajan roots, finished unbeaten on 58 from 36 balls alongside Salt, in the process becoming the youngest man to hit a T20 fifty for England, aged 21.
England pulled off the highest T20 chase at the Kensington Oval with 19 balls still to go, after West Indies earlier made their imposing total in remarkable fashion, reaching 183-9 despite falling to 18-3 and 117-8.
They managed a further 65 runs in the last 5.2 overs after the eighth wicket fell, with number 10 Gukadesh Motie flogging 33 from 14 balls.
That came as England lost their grip despite a strong start to the innings, with Saqib Mahmood taking 4-34 and Adil Rashid 3-32.
The second of five T20s begins on Sunday at the same ground.
Player of the Match, Salt said: "This is probably the place I'm happiest. I enjoy batting here. I've grown up on these surfaces.
West Indies captain Rovman Powell: "It didn't really come off. Credit has to be given to the boys. We still managed to score 180. We have highlighted the way we want to play. In the bad times we have to keep backing ourselves."